So are you asking if they are involved with each other? Or is more of you want to know what each one does lore wise? Like the role within society etc? I made a topic about are the black mages the warlocks of eorzea(spoiler warning in the topic). Thaumaturges use their inner aetheric energy to cast spells while black mages instead suck the aether out of the atompshere to cast stronger destructive spells. Plus thaumaturgy/black magic is connected to the void the same void where all the demons live.
In the topic someone explained the duties of the thaumaturge and lets just say its pretty dark and black mage is even worse considering what black mages will have to deal with in their daily life. They both use the dark arts but black mage is even darker. Here if the topic if you want to read it.
http://forum.square-enix.com/ffxiv/t...ocks-of-eorzea
Now what I personally know on conjurers/white mages they are essentially druids( I haven't done their class quests I am just going by what friends have told me and what we see of them in the story). They deal with and heal the forest etc and commune with the elementals. Considering how they cast their spells since they work with the forest they would not like black mages one bit.
Since we suck the aether out of the atmosphere(which is not environmentally friendly compared to the other spell schools) to cast spells and since aether flows in living things I bet casting our major spells causes some trees or flowers to die around us. XD
Since I leveled arcanist to 47 I got a good understanding of them they are pretty much the normal mages of the game when it comes to casting spells. They see the patterns in the aether and convert this into logic/maths terms and they use that to cast their magic. So in simple terms just think of it as maths magic.Their spells seem like dark magic but its not that belongs to the thaumaturges/black mages.
Now onto their duties they are hardcore border control they deal with smugglers and the beastmen who live in limsa lominsa to ensure all their papers are up to date.
Well this is all I know I hope I answered your question. :X