The Final Fantasy III remake makes a few changes to the original game, the biggest being a cast of distinctive characters rather than the (mostly) blank slate Onion Knights. However, due to the nature of the remake, it's only just barely an improvement.
Evidently, this was not always the case.
Within the code of the game, text strings remain from an earlier iteration of the game. These strings are extensive, as almost every area has at least one line that was cut from the game.
But not only that...
In the interview at the back of the DS version's strategy guide, it is mentioned that there were a slew of abilities that got scrapped from the game.