The European and Korean versions of the game uses the US dub for all of its languages (all further localized Layton games starting from the second one have a different dub for Korean language and each European language), with a single difference: all the lines said by Luke have been re-recorded by a native British voice actress. The British accent imitation done by the American actress was reportedly borderline offending for the UK play testers. This change was carried in all the Layton UK versions.
One of the puzzles in the game (Puzzle 129: Four Balls), the initial position of the blocks was changed in the European and Korean versions so that in the European and Korean versions 27 moves are required to solve the puzzle rather than 28 moves. This was changed because the original distribution resembled a swastika. However, the text for the solution wasn't changed and still states that the solution to the puzzle requires 28 moves. This change is also present in the HD re-release.
While the US and Japanese versions of the game give you the same amount of Picarats for each of the puzzles, a few puzzles give out different Picarats in the European and Korean versions. This change in the amount of Picarats obtainable affects the maximum Picarats that can be obtained between versions. In the US and Japanese versions it's 5312 Picarats, while in the European and Korean versions it's 5302 Picarats.