As far as BLM goes, it branches up from Thaumaturge. Using the elements of Ice and Fire, the elements draw directly on the land's aether. Ice draws on it and gives it to the BLM for the BLM to then channel through fire, fire aspect using the aether of the land to make hotter, more destructive fire. It's supposed to be bad for the land, iirc. The techniques within the BLM soul crystal are only to use the elements for purely destructive ends.

In FFXIV we're a goody two shoes, so we're just really powerful at magic. In the quests you do sometimes close void portals that other, nefarious BLMs open... usually by blowing them up.

The three elements that BLM wields are a direct nod to BLM from Final Fantasy(1987). In that game the three offensive elements that didn't involve instant death were Fire, Ice, and Lightning.

AST draws aether from heavenly bodies, and then manipulates it to restorative magic as well as to empower their cards. Basically it only differs from conjury or white magic by the fact that those draw exclusively from nature/the mage, whereas AST draws exclusively from other planets, with a few exceptions, like Earthly Star and drawing from the planet beneath your feet.

That's like, the cliff notes versions. I'm sure they're not 100% spot on, but I'd have to go rewatch AST cutscenes to remember properly.