What does powering up actually do?

Increases your CpM, which is key to the amount of damage you do, and take.
GP explain it all here...

https://pokemongo.gamepress.gg/damage-mechanics

https://pokemongo.gamepress.gg/cp-multiplier

Basically, when everyone talks about dps...it is actually 'power per second' (pps) that gets multiplied by a factor determined by your level, which will give the actual damage done.

To answer your question, a mon at 2000CP will do more damage, and mitigate more damage taken than the same mon at 1500CP.

Powering up raises a pokemon half a level, which then causes its attack, defense, and stamina to be recalculated, which then causes its CP and HP to be recalculated.

To put it in yet another way, the attacker level vs the defender level is a factor in the equation that also determines how much they damage each other. So powering up raises your level, improving the ratio to his level. A better ratio reduces your damage taken and increases the damage your giving.