Well Final Mix (the KH1 one) isn't story-changed or something else.
The thing is with the Organization that you can fight Xenmas and the Optional Bosses Sephiroth and Ice-Titan, which weren't presented in Original KH (Japan)
You get an ansem report (i think there are 12 or 13, have to look again) and you get some new keyblades, which are diamond dust and one winged angel
new enemies and old ones new, mostly like clowns, colored
well you see a sequence with riku on the dark side etc...
well thats all i think story's the same
The Final Mix Productions are called Final (maybe) cause these are the FINAL changes of the game and the most important.
Kingdom Hearts Final Mix, RE:CoM and KH2 Final Mix are the "true" Storyline and "true" Kingdom Hearts
Since it's traditional to KH you can call the Old Games just Beta Versions which are good enough to sell!