I think the worst part about the DS remake is that the dungeons go on forever, there's no save points outside of like inns, and there's no easy way to get /out/ of dungeons either. Maybe the original was the same but that's the kind of thing you really need to fix in your remakes, especially if they're on a handheld.
Like here's the really awesome thing about Shining the Holy Ark, for comparison -- dungeons still don't have save points, but they are usually divided up into sections locked off by key items, and once you get those key items you open shortcuts. So there's this flow where you play a bit -- I'm in the South Shrine now. Wonderful music and atmosphere btw -- find some item that lets you progress further n the dungeon, and then you can warp back to town, heal up, and save, and when you get back into the dungeon there's minimal backtracking because the thing you got opens new paths.
... A lot of more recent RPGs still don't get this right >:T
Yeah, they're identical to the NES/Famicom version unfortunately. At least the buff to the White Mage's spell pool helps this somewhat, though buffing the enemy encounters (and bosses) across the board to compensate for the fights against more than 3 enemies is somewhat more debatable (makes the Ancient Cave more manageable, but the rest of the game doesn't benefit nearly as much; OTOH, nerfing the Ancient Cave alone is almost worth the price of admission).
As for escaping dungeons quickly, both versions of FFIII give you the Exit and Warp spells (though they're both set to spell levels that have better things to do than to escape dungeons; spell balance is so fucked up in FFIII, Exit really should've been either a Level 2 or 4 WM spell).