Adaptation Displacement: Seeing as the main characters of the NES version were blankslates, it's no wonder that most fans nowadays see Luneth as the face of III.
Demonic Spiders: Every enemy that splits when it takes physical damage from a non-katana. They can quickly prove overwhelming if you're not prepared for them.
These foes are easier in the DS version, in which these enemies only split on their turns if they're hit with a non-katana instead of immediately after they're hit, and only three of them can be out at a time, making it easier to kill them with a multi-target spell, or with the Darkness Wave of a Dark Knight.
Game Breaker: The Onion Knight class, in the DS remake.
Good Bad Bugs: A bug involving the inventory in the Famicom version allowed the player to acquire the very powerful Onion Equipment at the start of the game. (It takes a huge amount of time to set up, however.) Another one in the DS version allows players to duplicate any consumable item in the game, up to and including the rare and precious Phoenix Downs.
Needs More Love: This game in general. You'll notice that if you look at all of the rereleases of the first six Final Fantasy titles to Keep Circulating the Tapes; this game has had the least rereleases.
It's odd, since in Japan, it was voted the fourth most popular FF game (after X, VII, and IV). Perhaps the No Export for You induced apathy for the game in other countries is why Square doesn't pay it as much mind, as it wouldn't be profitable outside of Japan.
Which doesn't make sense. How can the west feel apathy if we never played it? It's just that a Wonderswan FFIII remake never happened. That's all. Square was too lazy to put it on GBA during the Finest Fantasy for Advance campaign.
Nightmare Fuel: The Nepto Dragon's design in the original Famicom version is... creepy.
Seinfeld Is Unfunny: The class system this game introduced to the series has been done better in so many other games by the time it was finally localized it doesn't reach the standards of even V. Never the less, several gamers found it to be a refreshing break from the immense complexity of other titles in the series.
That One Attack - In the DS remake, Cloud of Darkness will use an attack called Particle Beam, dealing massive damage. In the original, it's all she ever uses.
She does have one alternative attack in the NES version, but it's arguably worse than Particle Beam (or "Flare Wave" as the NES version called it), because it's a physical attack that always hits and usually deals 9,999 damage points.
That One Boss: Salamander, Echidna, Ahriman and Garuda.
Woobie, Destroyer of Worlds: Xande. The guy used to be an immortal sorcerer who had his immortality taken from him while Doga and Unei got more power than ever. You'd probably be a bit upset too.