I have a Canon SX100IS and a Olympus Tough 8000. Both are pretty compact, the Tough 8000 is smaller.
The Canon has notably better pictures. Better lens = sharper, and also better colour in the images. The Olympus tends to make everything look slightly blue.
OTOH the Olympus is smaller, lighter and tougher. I take it canyoneering and river rafting and like that and it takes shots that are "good enough". It doesn't freeze at -20C and you can drop it off a cliff into a pool and it still works.
Basically the best camera is the one you take the pictures with. A camera that is heavy and bulky might take good pics but if it's not in your hand or close by when you see that awesome shot then you won't be taking the picture. For climbing shots that means something you can climb on lead with and not worry about breaking.
Forget megapixels and look at image quality. Once you have quality get the best zoom you can in a compact camera and you should be golden. Finally don't pay more than $300 unless you think you're going to seriously pursue commercial picture sales. You can get a lot of good camera for $300 nowadays. And you aren't out too much when you forget it at the bivi and hike out.