Since Olivier has completely excised R&G, this sequence has gone with them. We lose many key lines and one of the best known speeches, and that certainly has an effect on the play. One might argue that Olivier plays clinical depression (i.e. melancholy) well enough that we hardly need the speech describing his mood, and indeed, Olivier's adaptation is more dour for losing the witty banter between Hamlet and his school chums. We are left with a single line, spoken to Polonius when he comes to announce the players: "He that plays the king shall be welcome." Olivier obviously thought it important, and it is. It ironically sends up Claudius by replacing him with a more welcome pretender.

But however little the effect on the particulars of the plot, one still misses the wit and poetry of Shakespeare's words.

In Branagh's production, R&G arrive on a small steam train, evoking their childhood friendship with Hamlet. Almost immediately, they fall into the sort of clever banter they used to indulge in with the witty prince. The setting also seems important. Meeting them outside Elsinore reminds us that they are outsiders to the Court they desperately want to be a part of, but exteriors also give weight to Hamlet's line about being a king of infinite space, and in his black figure against the snow connects Hamlet to the shadowy dream (his unrealized revenge) and the outstretched echo of a hero or monarch who is but a beggar (as he refers to himself soon after). Another reason to place this scene outside is to reveal R&G's duplicity. We have already seen them inside Elsinore, so their surprise "arrival" is a con. When Hamlet tells them their news is not true, it seems at first that he means "the world's grown honest", but it may just as well refer to their entire arrival.

Branagh plays Hamlet as emotional here. As soon as he calls R&G on their untrue news, his eyes start to water, perhaps in anger. He realizes early on that he has been betrayed by childhood friends. This emotion pours out when he confronts them about it. And he does so head on. "I have an eye of you", scripted as an aside, is here a warning. Hamlet in fact uses his emotion to shame them into admitting their connivance with Claudius. He then seems to take them into his confidence, the lie out of the way so to speak, but he may be playing with them. Branagh plays him as sometimes vulnerable, sometimes mad, sometimes angry, giving them knowing, ironic looks. For example, he starts "What a piece of work is a man" while looking at Guildenstern, an edified man he will bring down in the course of the speech. He also looks at each of his friends in turn when he mentions the hawk and the handsaw, playing on the joke that the characters are nearly identical, but also saying that he knows friends from enemies.
For their part, they don't quite know what to make of him, and where Horatio seems to be on the same page as the prince, their interactions are often inappropriate. They are OLD friends in a very real sense (as opposed to LONG-TIME friends). They are only comfortable in nostalgia. They can handle banter and theatrical gossip, but they can't deal with Hamlet's depression or flights of poetry. This culminates in their turning one of the play's key speeches into a gay joke, their chuckling visibly insulting Hamlet. They revert to childhood when interacting with their childhood friend.
The theatrical gossip section is not often played, and Branagh wastes as little time as possible, making the rapid-fire discussion so quick it's hard to really get any meaning out of it. So why is it there (in the text)? It of course tells us something about R&G and Hamlet. Thematically, they're also talking about pretenders. Young actors coming on the scene and stealing the work of more established ones, their worthiness in question. Hamlet relates this to his uncle usurping the throne - and getting the two suck-ups to laugh along with his joke - and condemns "fashion" for elevating both. And then there are the pretenders that are R&G, ironically trying to elevate themselves like the "boys" did. So while Branagh does not necessarily convince us that this section needs to be in play, it does support many of the play's ideas.

In the next sequence, Rosencrantz & Guildenstern finally catch up to Hamlet. Though in the guise of old school chums, they are really there at the King's behest to spy on the prince. This does not mean they have to be sinister figures. They may, in fact, have Hamlet's best interest at heart (as does the Queen who first sent for them), but they do find themselves at cross-purposes with Hamlet. It is perhaps telling that their conversation is not in blank verse. Scenes with Horatio, who has a similar relationship with Hamlet on the surface (school friends), are still in verse. Do R&G not rate the heightened language of the Court? Where Hamlet treats Horatio as an equal (and at times, even a better), he talks down to R&G, bringing himself to their level and basically running circles around them. It may be interesting to note that when R&G do speak in verse (to the King and Queen, for example), they do so at their own peril, paranoid and halting in their delivery. This is a world they aspire to, but which is forever out of reach. Hamlet has outgrown them.

The sequence also contains the quotable "What a piece of work is a man" speech. As with the previous sequence, here's a link to a website that offers video comparison between more Hamlets than I'm doing. You can compare yourself. But first, lets look at the text itself, in italics to differentiate it from my own comments:

Enter ROSENCRANTZ and GUILDENSTERN

LORD POLONIUS: You go to seek the Lord Hamlet; there he is.
ROSENCRANTZ: [To POLONIUS] God save you, sir!
Exit POLONIUS

GUILDENSTERN: My honoured lord!
ROSENCRANTZ: My most dear lord!
HAMLET: My excellent good friends! How dost thou, Guildenstern? Ah, Rosencrantz! Good lads, how do ye both?
ROSENCRANTZ: As the indifferent children of the earth.
GUILDENSTERN: Happy, in that we are not over-happy;
On fortune's cap we are not the very button.
HAMLET: Nor the soles of her shoe?
ROSENCRANTZ: Neither, my lord.

This sums up the duo's place in Denmark. They have a tenuous connection to the Royal family, but have not yet managed to rise to inner Court which the aspire to. They'll go on to jealously wonder why Hamlet, who to them, is at the very top of the social scale, feels he is at its bottom.

HAMLET: Then you live about her waist, or in the middle of her favours?
GUILDENSTERN: 'Faith, her privates we.
HAMLET: In the secret parts of fortune? O, most true; she is a strumpet. What's the news?

It amuses me that this coarse joke has been removed from my pocket copy of the play, published for use in the school room. Beyond the joke itself, it is telling that Hamlet considers Fortune to be a whore who raised his murderous uncle to the loftiest position in the land, and in so doing, makes a statement about his own mother who let Claudius climb on top of her. It continues Hamlet's motif of linking adultery and sexual sin to the female principle.

ROSENCRANTZ: None, my lord, but that the world's grown honest.
HAMLET: Then is doomsday near: but your news is not true. Let me question more in particular: what have you, my good friends, deserved at the hands of fortune, that she sends you to prison hither?
GUILDENSTERN: Prison, my lord!
HAMLET: Denmark's a prison.
ROSENCRANTZ: Then is the world one.
HAMLET: A goodly one; in which there are many confines, wards and dungeons, Denmark being one o' the worst.

The reader may well find that this line foreshadows the final fate of R&G, so long as one imagines a short stay in a dungeon before their execution.

ROSENCRANTZ: We think not so, my lord.
HAMLET: Why, then, 'tis none to you; for there is nothing either good or bad, but thinking makes it so: to me it is a prison.
ROSENCRANTZ: Why then, your ambition makes it one; 'tis too narrow for your mind.

I've often found that people ascribe their own motivations to others. Those who act in bad faith assume bad faith on the part of others, and so on. How R&G interpret Hamlet's words tells us more about their own thought process than Hamlet's. They are motivated by ambition; he is not. It does beggar the question as to whether they believe Hamlet should be on the throne by filial right. Are they fishing for answers and testing their own theory for Hamlet's distemper? And if that is their belief, the characters would be trapped between a rock and a hard place trying to please both the pretender King and the would-be King, so as to stay in both's good graces. Their death sentence is signed not by their choosing the wrong side (they don't know they're carrying orders to have Hamlet put to death), but for hedging their bets and not choosing a side at all (except their own).

HAMLET: O God, I could be bounded in a nut shell and count myself a king of infinite space, were it not that I have bad dreams.
GUILDENSTERN: Which dreams indeed are ambition, for the very substance of the ambitious is merely the shadow of a dream.
HAMLET: A dream itself is but a shadow.
ROSENCRANTZ: Truly, and I hold ambition of so airy and light a quality that it is but a shadow's shadow.
HAMLET: Then are our beggars bodies, and our monarchs and outstretched heroes the beggars' shadows. Shall we to the court? for, by my fay, I cannot reason.

One of two connections made between beggars and kings in the play. Hamlet is consistently deflating his own royalty and that of others. It's part of the fatalism of the play - we all return to the earth in the end - as well as a slight to the current King - if he can be king, then anyone can. This particular line mirrors R&G's own ambitions, the shadows/dreams being their own. Hamlet likens them to beggars, which they are. They beg for attention and power.

ROSENCRANTZ GUILDENSTERN: We'll wait upon you.
HAMLET: No such matter: I will not sort you with the rest of my servants, for, to speak to you like an honest man, I am most dreadfully attended. But, in the beaten way of friendship, what make you at Elsinore?
ROSENCRANTZ: To visit you, my lord; no other occasion.
HAMLET: Beggar that I am, I am even poor in thanks; but I thank you: and sure, dear friends, my thanks are too dear a halfpenny. Were you not sent for? Is it your own inclining? Is it a free visitation? Come, deal justly with me: come, come; nay, speak.

Again, Hamlet confuses a Royal for a beggar. In this case, himself. The theme of this sequence will eventually bloom in "What a piece of work is a man" in which Hamlet

GUILDENSTERN: What should we say, my lord?
HAMLET: Why, any thing, but to the purpose. You were sent for; and there is a kind of confession in your looks which your modesties have not craft enough to colour: I know the good king and queen have sent for you.
ROSENCRANTZ: To what end, my lord?
HAMLET: That you must teach me. But let me conjure you, by the rights of our fellowship, by the consonancy of our youth, by the obligation of our ever-preserved love, and by what more dear a better proposer could charge you withal, be even and direct with me, whether you were sent for, or no?
ROSENCRANTZ: [Aside to GUILDENSTERN] What say you?
HAMLET: [Aside] Nay, then, I have an eye of you.--If you love me, hold not off.
GUILDENSTERN: My lord, we were sent for.
HAMLET: I will tell you why; so shall my anticipation prevent your discovery, and your secrecy to the king and queen moult no feather. I have of late--but wherefore I know not--lost all my mirth, forgone all custom of exercises; and indeed it goes so heavily with my disposition that this goodly frame, the earth, seems to me a sterile promontory, this most excellent canopy, the air, look you, this brave o'erhanging firmament, this majestical roof fretted with golden fire, why, it appears no other thing to me than a foul and pestilent congregation of vapours. What a piece of work is a man! how noble in reason! how infinite in faculty! in form and moving how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals! And yet, to me, what is this quintessence of dust? man delights not me: no, nor woman neither, though by your smiling you seem to say so.
ROSENCRANTZ: My lord, there was no such stuff in my thoughts.
HAMLET: Why did you laugh then, when I said 'man delights not me'?
ROSENCRANTZ: To think, my lord, if you delight not in man, what lenten entertainment the players shall receive from you: we coted them on the way; and hither are they coming, to offer you service.

As part of the various inversions in this section of Scene 2 is the change of tone taking us from Hamlet's sublime and fatalistic speech to R&G laughing at a gay joke. This is another sign that R&G are just on Hamlet's level, intellectually or poetically. It also acts as a bridge for a conversation about Elizabethan theater. Hamlet is very much about a man putting on a play, and from here to the play within a play, it also becomes about theater itself. This section (along with the later speech to the Players) gives us an idea of what Shakespeare's theater was like, but also tells us there was a lot of what he considers bad theater going on. The sequence starts with Hamlet making an ironic comment in the context of his uncle's usurpation of the throne:

HAMLET: He that plays the king shall be welcome; his majesty shall have tribute of me; the adventurous knight shall use his foil and target; the lover shall not sigh gratis; the humourous man shall end his part in peace; the clown shall make those laugh whose lungs are tickled o' the sere; and the lady shall say her mind freely, or the blank verse shall halt for't. What players are they?
ROSENCRANTZ: Even those you were wont to take delight in, the tragedians of the city.
HAMLET: How chances it they travel? their residence, both in reputation and profit, was better both ways.
ROSENCRANTZ`I think their inhibition comes by the means of the late innovation.
HAMLET: Do they hold the same estimation they did when I was in the city? are they so followed?
ROSENCRANTZ: No, indeed, are they not.
HAMLET: How comes it? do they grow rusty?
ROSENCRANTZ: Nay, their endeavour keeps in the wonted pace: but there is, sir, an aery of children, little eyases, that cry out on the top of question, and are most tyrannically clapped for't: these are now the fashion, and so berattle the common stages--so they call them--that many wearing rapiers are afraid of goose-quills and dare scarce come thither.
HAMLET: What, are they children? who maintains 'em? how are they escoted? Will they pursue the quality no longer than they can sing? will they not say afterwards, if they should grow themselves to common players--as it is most like, if their means are no better--their writers do them wrong, to make them exclaim against their own succession?
ROSENCRANTZ: 'Faith, there has been much to do on both sides; and the nation holds it no sin to tarre them to controversy: there was, for a while, no money bid for argument, unless the poet and the player went to cuffs in the question.
HAMLET: Is't possible?
GUILDENSTERN: O, there has been much throwing about of brains.
HAMLET: Do the boys carry it away?
ROSENCRANTZ: Ay, that they do, my lord; Hercules and his load too.
HAMLET: It is not very strange; for mine uncle is king of Denmark, and those that would make mows at him while my father lived, give twenty, forty, fifty, an hundred ducats a-piece for his picture in little. 'Sblood, there is something in this more than natural, if philosophy could find it out.

Are we to understand that Shakespeare, at this point in his career, felt the sting of competition? It's the same today: Things that are popular are not necessarily good art. It's what makes people hide behind the idea that quality is all subjective, when clearly, it is not. At the end of this oft deleted conversation, Hamlet brings things back to his situation, comparing Claudius to bad theater, popular but lacking in quality.

Flourish of trumpets within

GUILDENSTERN: There are the players.
HAMLET: Gentlemen, you are welcome to Elsinore. Your hands, come then: the appurtenance of welcome is fashion and ceremony: let me comply with you in this garb, lest my extent to the players, which, I tell you, must show fairly outward, should more appear like entertainment than yours. You are welcome: but my uncle-father and aunt-mother are deceived.
GUILDENSTERN: In what, my dear lord?
HAMLET: I am but mad north-north-west: when the wind is southerly I know a hawk from a handsaw.

At the very end, Hamlet lets slip something of the truth. It may not be right to say that he lets down his guard exactly, because the comment, at least to R&G's ears, is coded. It is not the only time that Hamlet sounds mad even as he tells us he isn't.

Halliday's Fishmonger Scene does away with Polonius entirely and makes it about Hamlet's reading, and in so doing gives us a reflective, personal song that intersects with various ideas in the play. You can listen to it HERE, and I follow the text with a rough translation.

Je lis
Quand la vie est jolie, jolie, je lis
Quand il fait triste, et il fait gris, je lis
Quand je suis seul dans mon lit, je lis
Quand même les rêves me fuient, je lis
Quand je cherche un coin d’oubli, je lis
Quand j’ai besoin d’un alibi, je lis
Quand on me sonne l'hallali, je lis
Quand je perds le goût de la vie, je lis
Quand je lis, je lis, je lis quoi ?
Des mots…

I read
When life is pretty, pretty, I read
When it is sad, and it is gray, I read
When I'm alone in my bed, I read
When even my dreams desert me, I read
When I look for a forgotten corner, I read
When I need an alibi, I read
When they trumpet the kill, I read
When I lose the taste for life, I read
When I read, I read, I read what?
Words...

The short song has Hamlet reading, rain or shine, homogenizing his life which he will later compare to a prison. Most of the lines speak to his fatalism, undercutting the hopefulness of the first in which the near homonyms jolie (pretty) and je lis (I read) are playfully confused. The song could almost be sung with jolie (pretty) in place of je lis (I read) throughout, extending the irony. Is Ophelia on Hamlet's mind still? Is his loneliness (colored by his reading books, to use Polonius' words in a later scene) making his mind wander from the page to the love of his life? There may lie the contempt one might understand from the "punchline". I'm reminded of the phrase Jonathan Swift wrote on the envelope containing a lock of his dead wife's hair, "Only a woman's hair". The words Hamlet reads are meant to be a refuge, but in no way do they replace the relationships he has broken off. Without the comic context of the Hamlet-Polonius interplay, "words, words, words" takes on some of the meaning behind the "What a rogue and peasant slave am I" speech, in which Hamlet is outraged that he must "unpack his heart with words", or the earlier speech on how his grief cannot be properly contained in "actions that a man might play".

The song reveals a possible staging where "words, words, words" can be pronounced in an aside to take on more personal meaning.

The original
It is difficult to convey the comedy of the scene, and the original Classics Illustrated doesn't even try, though perhaps it was removed to avoid getting into the fishmonger/pimp metaphor. So in this version, Polonius and Claudius conspire to send Ophelia to meet Hamlet, and though the "To be or not to be" speech immediately follows, their plan comes to fruition thereafter.

The Berkley versionTom Mandrake's adaptation retains the sequence, but makes deep cuts into it, ending it before Polonius asks Hamlet what he's reading. This removes much of Hamlet toying with Polonius - and all my favorite lines! This is a largely humorless adaptation, Mandrake's facial expressions and atmosphere better suited to anxiety and anguish. His Hamlet is deadly serious, and his Polonius a sad old man, bewildered and saddened by events. We lose nothing of the plot, of course, and has Rosencrantz & Guildenstern show up earlier, before Polonius can get even more insulted, which in this vision, sometimes seems unduly harsh.