What do "power" attacks actually do?

What difference do power attacks make?

I use double daggers and doing a power attack takes lots of stamina but doesn't seem to give me any advantages, other than a nice animation.

Are there any difference between 1 handed and 2 handed (duel wielded) power attacks?

Solution 1:

According to the Unofficial Elder Scrolls Pages, power attacks do double damage and have a chance of staggering the opponent.

Solution 2:

Aside from the increased damage/stagger chance, you can also get perks to further improve your power attacks, including such things as sweep attacks against all enemies in front of you (with 2-h weapons) or a chance to behead enemies (with 1-h weapons).