What's the diffrence between spells and abilities?

If it was only spell damage, it would be abilities that you cast on opponent. It includes ability too, for auto attack modifiers like Jax W, Poppy Q, Nasus Q.

If I recall correctly:

Ability = QWER

Spell = Summoner spells

Abilities are what are on QWER minus things that are on-hit effects or apply on-hit effects. Spells are abilities as well as summoners spells and active items.

I think it means any spell damage (i.e. magic damage), but also any ability damage (i.e. can include physical damage abilities).

A dev actually answered that somewhere at some point in a very summarized way, but basically the difference is only on the backend and has no reason to exist in the masteries. It's something along the lines of "Spells are QWER that actively do something, like most skills, and Abilities are AA enhancers, on-hit effects, passives, etc."

If you can cast it and there's an effect, it's a spell.