The reason why the designs are different is that FF5 was made for a 16-bit game system - Super Nintendo, I think. Back then, game programs couldn't recreate on-screen the character designs that Square artist Amano had sketched out for the FF5 characters - the graphics weren't good enough. So the game programers redesigned the in-game sprites so that they would work on-screen, changing Amano's ideas.

When FF5 was re-released for the PSX, technology had caught up, so the programers thought it would be nice to use Amano's ORIGINAL designs for the characters in the new FMV sequences. The CG cuts of Bartz, Reina, Galuf, Krile, and Faris were how the characters were SUPPOSED to look from the very beginning.

You can find some of the original FF5 artwork here: http://www.ffshrine.org/ff5/ff5_amanoart.php