The basics remain intact - plot, dungeons, bone-crushing difficulty - along with some minor tweaks that help smooth out some of the original's rough edges, such as making more character classes viable during the endgame. The original game has been completely rebuilt using vibrant 3D graphics along with a freshly remixed soundtrack.
Thankfully, it's also an excellent remake. The definitive version: Final Fantasy 3 (DS/PSP/PC/Mobile) As this is the only official English version of the game available in the West, it's the winner by default.