I've only played the Anthology version, but to my knowledge, the only differences are
1) the added FMVs - one at the begining and one at the end.
2) the translation - the ROM's is fan-made, whereas Anthology's is the "offical" translation. This is usually the big reason why people like the ROM version better; some of Square's translation is a little silly, like Faris' accent is a bit much and the butchering of names (i.e. SALSA?!). However, I never found it that much of a detriment to the game as a whole.

Anything else, like the character stats in the end, should be in both versions.

Therefore, my advice is, if you've already played Anthology and want to start a new game, why not try the ROM? Then you'd know the differences and be able to determine a preference for yourself.