I can't answer very specifically, since my Black Mage character is only level 30, but I can give some general thoughts as a fellow damage dealer.
I think some people get too fixated on the word "rotation" and get into the mindset that it's all about a single sequence of skills to use for all situations. While there are certainly skills that support each other and using them together in a certain order is important, a great deal of of your effectiveness comes more from your ability to recognize which sorts of situations call for which skills (or for which sequences of skills).
Anyway, as to why your effectiveness as a damage dealer is important to clearing dungeons, while speed is certainly one effect, there are indeed fights where failing to deal damage fast enough will cause a wipe.
You've actually come across this before quite a few times. Starting way back at level 20, when the nail appeared in the Ifrit fight, if you didn't kill that nail quickly enough, Ifrit would launch an attack that could wipe the whole party. Later, much the same thing happened with Titan's Heart. These low level DPS checks can sometimes be overlooked now, though, since most players are so overleveled and overgeared that a great deal of the original challenge is lost, and most players never see what would have happened if they hadn't done enough damage fast enough.
Post level 50 (and even more so once you reach endgame) you'll start finding DPS checks that are a lot more challenging than the ones seen while leveling that far, ones where you might not get by purely by overgearing everything, and will need some skill in using the most effective skills at the most effective time in order to destroy the metaphorical ticking time bomb before it blows up in your face.
They do different things.
Fire II is an AoE attack to all enemies near the one you targeted. Fire III is a single-target attack. If you're only facing one or two enemies, then Fire III is more powerful, but if you're facing a whole crowd of them (any more than two actually), Fire II is.
As an initial Fire-type skill, though, Fire III has the advantage that it immediately gives you the maximum 3 stacks of the Astral Fire buff, rather than having to build them up one by one and needing a transpose. Both Fire and Fire II, on the other hand, would start off just removing the Umbral Ice if Transpose isn't available, then next time adding a single Astral Fire, then a couple more times before you get up to 3 stacks of Astral Fire.
So if you're facing a group of mobs and need to switch from Ice mode back into Fire mode, you'd use a single Fire III to give you your maximum Astral Fire, then you'd switch to Fire II for AoE damage.