Bottom line is, you can define and divide up "kinds of Magic" however you see fit, what makes the most sense to you, or your setting/game... or, as illustrated by others here, makes the least sense, for that matter, if that produces the most fun and flavor for your group.
As I recall, Dragonlance had all mages tied to their alignment and corresponding "deity of magic/moon of a certain color." This affiliation, by the setting/playable material, dictated what types of magic you could learn/use. Like "Red [Neutral] Mages" were the only ones who could learn illusion spells, I think (s'been a long time since I looked at them). "White [Good] Mages" were the only ones who could learn abjurations, and so forth.
[sparked by the current Alignment conversation happening elsewhere] One could make a world in which all magic is simply divided up into Light or Darkness....and then it's up to the individual wizard to determine how far into one power source or the other they are comfortable or willing to delve.
My understanding of Magic:TG, admittedly limited, associates the different kinds of spell-users by thematic colors. "Blue" is mind magic, I think[?] I've read. "Green" magic controls plants/nature. Am I close? Some people seem to like that.
Elementalists have been around forever and since Avatar:TLA, I am certain setting after setting have made their magic all focused around the four cardinal elements. Plenty more worlds go for the classic Asian elements, and make it six or seven: plant and metal as separate from "earth," right?
I have an OSR clone (not sure which one off the top of my head, maybe [edit, I looked through my stuff]Fantastic Swords & Wizardry? Fantastic Heroes & Witchery[/edit]) that breaks all magic down into "White/Grey/[and of course]Black" Magic. Magic in this game/worlds of this system have your type of magic based in "Intention" moreso than alignment. Are you lookign for a spell that "does good/benefits" people or "harms/disrupts/furthers evil." Most "schools/descriptors" of different spells fit into one, two, or all three if these "categories" of magic. Your class dictates which type of magic. The basic "arcane magic-user/wizard/mage" class has access to all three, I believe.
You could make a world where the sorcery existed in the world before people...created by a set of Primordial Animals. You could break this down by "allowed schools" or by spell slot level, or by theme of the animal or whatever else you wanted. Your power in the world is determined by how many different animal totems you find/learn. If you know Lion, but never learned Dragon, then you can't cast fireball. All mages wander the world seeking out masters of various types and/or looking to accumulate the proper animal glyphs to pile up their power. You don't have "Illusoinists" and "Evokers" and "Necromancers" you have Bear wizards and Bat wizards and Griffon wizards.
An Elementalist "Fire mage" can hypnotise someone with their scintillating dancing colored flames and tell them what to do. The "Green mage" [druid] conjures and combines fragrances and spores to cloud the target's mind. The "Psychic" Enchantress just flat out telepathically suspends judgement and tweaks emotions to be agreeable.
All/any of them have "Charm Person" on their spell list. Charm Person is still an "enchantment" spell, specifically a "charm" as opposed to a "compulsion." But how the individual magic-worker achieves the same spell effect need not be the same from caster to caster...or even class to like class.
So, yeah....long story, long, the differences are whatever you want/make them to be. The schools have existed through the game. They are just there. A shorthand for categorizing and organizing the endless bevy of spells and magic that have NO boundaries or defined edges...because it's MAGIC. By definition anything is possible.
Did it have to be these eight schools? No, it certainly didn't. But in the roots of magical practices, legends, characters, and myths of the world, and based on the game origins whence this allll is spawned, they make sense...even if all the edges or where "this kind of magic" ends and "that kind of magic" begins are n't always crystalline.
If it is problematic or bothersome for you, take the illusion spells and enchantment spells and make them all one "Mind/Will/Electro-Chemistry of the Brain" Magic for your game world.