For the games I played multiple versions of, my favorites are

FFI: Origins. Same difficulty, new graphics, and taking my save with me between PS3 and Vita is awesome. Looks great on the latter's OLED screen.

FFIII - DS version, essentially a N64-esque remake that the game deserved.

FFIV - PSP. It captures the original we all fell in love with in the shiniest, most polished version. Giving players the option to switch back and forth between the enhanced and SNES original soundtracks was brilliant.

FFVI - Japanese rom, patched with the RPG One fanslation by Skyrender and ChrisRPG (RIP). FFVI is a completely different game than you think it is.