Fan translationFinal Fantasy III for the NES was never released by Squaresoft outside of Japan. However the game has been translated to English by amateur translators. In order to play this version you'll need a NES emulator and the patched/translated ROM file.
The game called Final Fantasy III (SNES), that was released in the U.S. was actually Final Fantasy VI (it was the sixth game of the series to appear in Japan).
InnovationsFinal Fantasy III was the first Final Fantasy game where you could summon monsters, meet moogles and where hit points are shown graphically above characters in-battle rather than as text in the caption box. It was the last Final Fantasy game to use the "level" magic system (from level 1 to level 8 spells).
MusicFinal Fantasy III is the NES/Famicom game that has the biggest quantity of songs in it. Technically, there are 58 tunes! (but some of them are just when the hero plays piano, etc...) The average NES games have 10-25 tunes in them.
RemakeFinal Fantasy III is a remake of this game, originally appearing on the Nintendo DS. It is the first version to officially appear outside Japan.
Information also contributed by Bregalad, Rodie and Steve Thompson