Wielding the power of google, I quickly turned up: http://amos.indiana.edu/library/scripts/papercuts.html
Essentially, one of the major problems is that they aren't clean cuts. This is the same reason why it's much more painful to cut yourself with a dull knife than a sharp one. But since paper cuts are so shallow, they heal over very quickly, and thus can easily trap irritants within them, which are often deposited in creating the cut in the first place. Combined with the fact that, as Red mentioned, paper cuts tend to occur on the fingertips which are very densely covered in nerve endings, and you've got a recipe for a painful minor annoyance.