[@Witch Cat] Well before I do the fully detailed version i think I can sum it up very briefly as this.

Specialization is the specific field of magic. They cover all the different kinds of spells you might have learned. All of the studies are super broad and one person can't learn everything from one without spending centuries on it. You're specialization might be a specific element you choose to focus on or how you make things in alchemy. It's in a sense your method of using magic.

Meanwhile the spells and abilities are single spells that you can cast or defined abilities you have. This would be where a shifter, who is a werecreature, would put their wolf form. This is where a Hunter would put their specific talents even if they aren't magical. This is where an Arcanist, who is a elemental mage, might put down a fireball , an ice spear, and a giant boulder spell.