Are there any software/game differences in the editions of Kingdom Hearts?
Final Mix is an updated version of the original which adds more content.
When the first game was brought to North American and European territories, they added more content in the form of enemies and cutscenes. Final Mix brings those updates to Japan as well, and added even more content on top of that.
The Limited Edition is the same game as Final Mix, but it includes extra physical goodies.
Ultimate Hits is a budget range by Sony; that is, those games are simply reprints released at lower prices.