What exactly is the relationship between the magic classes?

[Nialle]

ACN sees their magic as more 'math-y and scientific', THM is more 'raw destruction (and maniacal laughing)', and CONJ is more 'peacefully at one with nature and all life'.

At least that was my impression of the guilds.

[Catapult]

Indeed no conflict, but using the techniques of one school will have limited effect in another. For example, using Thaumaturgical techniques to move the earth would require a lot more energy and focus as it takes a great deal of power to move things, compared to making lightning dance across things. And trying too hard could kill you (ref: Sylphie).

PS: ARC = Archer. Arcanist = ACN.

[Enkidoh]

Also, as per my earlier post, each Guild actually serves a more social/mundane role in their respective nation, where their magical abilities are put to other uses other than warfare:

The CNJ's Guild is used to intercede between the Gridanians and the elementals on mundane matters, due to the treaty between the elementals and Gridania, the elementals insist that anything that would impact on the forest has to have the permission of the elementals first, and hence, the CNJ's fill this role, specifically CNJs especially sensitive to the elementals' whisperings who are known as 'hearers'. Accordingly, the CNJ's Guild also plays an important part in Gridania's government the Seedseer Council.
The THM's Guild are essentially funeral directors who developed their school of magic from their funerary rites (the guild receptionist clearly states this early on when you first join up), although a lot of Ul'dahns hold them in contempt as there have been rumours they demand elaborate funerals simply to graverob the deceased after the service.
The ACN's Guild are effectively the Customs house for Limsa Lominsa - although it's somewhat obscure exactly how they would use Arcanamia in this way, I presume it would be used to screen incoming goods for contraband without opening crates, or something. But I have no conclusive evidence in that regard so take that with a grain of salt.

As for Sylphie, as was stated she was using her own life force to power her healing spells, of which her spell potency seemed to be in equality with how much of her 'life' she used (hence why E-Sumi Yan expressly forbid her to use Raise - as Raise restores the (presumably recently) dead to life, if she used it it would have given life at the cost of her own.).

[Skies]

As for how arcanists work as a customs house.
Did you do the quests for it? It essentially has you running that work! (And they use carbuncles the same way one would use sniffing dogs)

[Enkidoh]

You have to remember though, when 1.0 was first released, classes were a lot different in how they functioned to how they are now (for instance, CNJ originally had the use of spells of all elements, including Fire, Lightning and Ice), and Jobs were not even in the game at all - they were only added towards the end of 1.0's life, hence classes had to be completely overhauled to compensate.

This is why THM lost Sacrifice, as it's traditionally a White Mage ability, and THM was made to upgrade to Black Mage, gaining the CNJ's Fire, Ice and Lightning spells (because CNJ was made to upgrade to WHM and focus more on healing). It had nothing to do with the lore of the class - it was purely to balance gameplay mechanics.

[Catapult]

Indeed, a great deal of the magic system application was retconned. It's probably important we don't read too much from the old system.