Do skills get more powerful each time they level up, or is it like Fallout where every 25 skill gains is the only time a boost happens?

Yes to both. There are additional perks every 25 levels, but there's a gain in weapon damage, or decrease in spell cost, or whatever as you go along.

Skills get better all the time, so your sword do more damage then your blade skill goes up, however every 25 skill level you get perks, somewhat similar to fallout 3 perks.
Weapons got power attacks; other skills often get features that can be passive like armor degrades slower, to new functions like invest in shops
Magic is most like fallouts lock picking and science every 25 level you get access to new spells.