Brave New World is the most popular mod, and it improves things nicely, but makes it more silly and changes dialogue, adds a few small scenes/npc's etc. The added functionality to the game is unparalleled, however---most importantly with the Esper Level system added, which is perfect and amazing. I would shamelessly recommend ReturnerScum's version of "FFVI:Is The Best Game Ever" mod, which is a more "serious" mod and, unfortunately, more difficult than Brave New World.
What I don't like about Brave New World:
-Sillyness, lame jokes, pointless immersion-breaking additions to scenes, npc's, etc. There isn't a ton of it but FF3 deserves better imho.
-Documentation could be a lot better. It has items, spells, but not much else if I remember right. No change history, tips, etc.
-Considering how much they did with the game, they didn't change the characters/items that much. Unique swords don't do anything special, for example.
-Game ends a lot faster, you can just breeze right through the World of Ruin
-Abilities are much more restricted than in the original. I think mods should expand playability, the original didn't overdo it.
What I Like Better About FFVI:ITBGE Mod
-Itemization is changed a lot and gets extreme, with dilemmas in your choices of equipment (+28 to stats or HP+100%?)
-You really feel like a poor rebel in the beginning, and you end up feeling like a rich hero in a wasteland by the end
-New spells are good and pretty
-Colosseum is boss
-Characters are very different than each other. Their abilities are more interesting and gives you more options, not less.
Drawbacks of FFVI:ITBGE Mod
-It just takes longer and when you're not in the mood, fuck that game. I still don't enjoy the 3 bosses at the end of the Airship.
-Esper leveling...so uncool compared to Brave New World
-You can't just breeze through the World of Ruin. It's a struggle until you gain at least 10 levels via story progression (5-10 hours). Since there's less story in WoR, it's more Skyrim, less Squaresoft, so you try out areas and whoops they're too hard, and you die. Good items are hard to find until you get to Phoenix Cave.
-The beginning is the hardest, because you're poor and you can really die if you miss a potion, get jumped by too many creatures, are a noob, etc.