Here is what I understand is different about Final Mix, it's a lot more than what's been said here so far:
- There are many new items and enemies, also, most of the enemies undergo a palette shift.
- There are several new scenes and a new battle with a guy called Enigmatic Man.
- There are a couple new modes of play.
- There's a new bonus ending video --- Deep Dive: Another Side Another Story. (This video is available for download at many sites.)
- Jiminy's journal has been updated --- once you've gained the final information for an entry, the king's seal (Mickey) appears next to the entry.
- There are new minigames involving the gummi ship.
- Various fixes --- changes in the locations of the 101 Dalmatians, the Red Trinity is moved out of the later-destroyed Oogie Boogie manor, etc. etc.
- There are three new Ansem Reports.
I found out all of this over at the Final Mix Changes FAQ (JIS) at
http://www.gamefaqs.com/console/ps2/game/516587.html a while back. If you want the full details, you should check it out. It's a lot more than just a few extra keyblades.