In a technical sense, Final Fantasy VI offers very few major innovations over its predecessors. The most obvious is a significant improvements in character graphics, primarily accomplished by using the same intricate two-dimensional sprites for the characters in both combat and in the outside world. This allowed for plot and character developments to take place during combat - when one character uses thought-to-be-extinct magic for the first time, another character reacts with shock, bouncing around the battle screen. But this is primarily an improvement based on the creativity of the designers - more time working with the harder limits of the SNES cartridges allowed more creativity within the form.
Final Fantasy VI also added customization within the game system. Just like FF4, FF6 has several characters in rigidly defined classes. Locke is a Thief, and the only character with the Steal command, whereas Strago is a Blue Mage who learns and uses monster's skills. However, each character can also be equipped with "Espers," summonable creatures who also teach magic skills and improve certain stats when the characters level up. This would provide the model for most future Final Fantasy games, as well as games from other companies: the characters each have their own personalities and are rigid when acquired, but can be developed in many different directions.
Perhaps more importantly, Final Fantasy VI offers massive improvements in thematic elements of the game. The rough edges of the setting are smoothed into a consistent whole. The coexistence of Tolkein-esque dwarves with science fiction Moon-dwellers has become a thematically consistent steampunk setting. This mixture of science fiction and fantasy was common in early RPGs, as we've seen. The early games in Final Fantasy have D&D settings with random airships, just like Wizardry and Ultima. But in Ultima, those previous embarrassments were ignored in future installments. In Wizardry, that artificial weirdness was maintained, but it was never a major focus. Final Fantasy reveled in its setting, and built its own style of science fantasy world, where airships and magic coexist.
In FF6, that setting drives the storyline, instead of the other way around. The evil empire in this game is using corrupting pure forms of magic in order to power their war machines to take over the world. The plot of the game derives naturally from the setting, and by and large, the major characters fit into both effectively.
I say "major characters" instead of "hero" or "protagonist" because Final Fantasy VI may well be unique in gaming history in that there isn't a character who fits that description. The first character we meet, Terra, is half-human, half magical Esper, and initially seems to be the protagonist, but is soon matched and eventually overshadowed by others. Locke the Thief is perhaps the most cliched, as the lovable rogue in the Han Solo mode. On the other hand, Celes, a general of the evil empire who turns against them, demonstrates much more depth, both tragically and comically, than her rough equivalent Cecil from FF4. Edgar is the king of a small nation which is trying to resist the encroaching empire, while Cyan was a knight of a kingdom which resisted and were all slaughtered, except for him. There are a few characters who don't entirely fit in the plot consistently, particularly some of the more gimmicky later characters like Umaru or Gogo, but one of the game's greatest achievements is how its most important characters are intertwined with the world and the plot. The Town-Dungeon-Boss conceit still exists, but it works organically with the rest of the game, instead of acting as an arbitrary challenge towards continuing the game.