FBC is so entertaining because of its high technical and artistic merits. Technically, the animation and artistry is hard to match even today. The imaginitive settings are gorgeously rendered, and the shear number of places the story goes, like a giant solar tower or a flying fortress or rocket-come-house, will keep you entertained and amazed.
While the character designs are decidedly muted, it allows for truly comprehensive and kinetic animation. Each character is defined more through the detailed animation of their little actions and quirks than anything they say. Watch the amazing gymnastic feats of Conan (titular boy of the future), and you will be as impressed as the more technically advanced action shows of modern times.
Of course, it's unfair to both "Future Boy Conan" and modern series to make comparisons between the two. FBC aims for much different goals than most modern action or drama shows. Specifically, FBC is less focused on its characters (each one more archetype than flesh and blood) and more plot and adventure oriented. Not to say the characters aren't entertaining, but don't expect real connections with any.
"Future Boy Conan's" real strength is its charm. If you have seen any Studio Ghibli films, you will be familiar with Miyazaki's trademark whimsy present through all of FBC. The entire series feels like a classic Ghibli film extended into a 26 episode series. For any fan of animation and adventure, that is a good thing.
I cannot recommend this series enough. "Future Boy Conan" is a real gem among animation. With child-like exuberance, the story never fails to engross you in its drama, but still maintains a kind of maturity in its animation and direction that makes it accessible to all audiences.