Wool Socks
Heat retention. Because of the way wool fibers are crimped together, it has greater bulk than other fibers which allow it to retain heat. If your feet are often cold, or you live a in a cold climate, wool socks may be a better fit.
Moisture wicking. Wool fibers are both hydrophobic (repels water) and hygroscopic, which is a fancy way of saying that it can absorb or give off moisture. In fact, they - wool fibers - can absorb up to 1/3 of their own weight in moisture before feeling "wet".
Static resistant. The fact that the moisture that it retains prevents static electricity from building up isn't that important of a selling point, but it is nice to know your feet won't get shocked when you slip on socks fresh out of the dryer.
Low-maintenance. This may sound gross, but wool socks don't necessarily need to be washed after each wearing. If they are left to air out, or spot cleaned, they can be worn a few times before being washed.
High-maintenance. Yes, we realize this contradicts the point above, but once you go to wash your socks, wool requires special care. Ideally, wool socks would be hand-washed in water with very little soap (soap, heat, and friction are wool socks' mortal enemies) and then left out to air dry. Machine washing wool socks is a bad idea!
Cotton Socks
Breathable. Cotton socks will be cooler than wool socks, which tend to insulate heat.
Machine washable. Unlike wool socks, cotton socks are fine for machine washing, which is one less thing to worry about.
Soft. Cotton is a very soft, natural fiber and will not itch the way cheap wool socks will. We add "cheap" because merino wool is a very fine, soft grade of wool that does not itch.
Choice. Cotton socks, for better or for worse, sell better than wool socks so manufacturers tend to make more cotton socks. More solids. More fancies. More options.
Allergic reactions less likely. Some folks are allergic to wool; fewer to cotton.
Links
http://www.meafordwool.com/? Look for the very special Qiviut-Socks !