Paper cuts are actually pretty subtle phenomena and, as you point out, sometimes you get them and sometimes you don't.

The first thing that's needed to break skin is pressure. Pressure is force per unit area. To get an idea of why it's pressure that matters more than force, think of what's worse -- someone standing on your hand with a stiletto heel (small area in contact) or a flat shoe (large area).

With paper, the area of the edge that cuts you is tiny, so even a small force will give a large pressure. This is not the whole story, however, since paper is floppy and can buckle before it manages to cut through skin. To get a paper cut you need to have the paper supported in such a way that it tends not to bend easily. This can happen when you have a book or a ream of paper where one page can easily slide out a little from the rest and present a cutting edge, but still be held tightly against bending.

Finally, skin is quite strong against direct pressure perpendicular to its surface, but fairly weak against shearing forces that are along the skin. This means that you need to drag the sharp edge along your skin to get a good cutting action. You've doubtless noticed this characteristic of skin with conventional cutting edges such as knives -- you can push quite hard against a blade without being cut, but would be sliced if you dragged your hand along the blade.

Paper cuts are often shallow so they don't bleed much, but they can be long and activate a lot of pain sensors in the skin so they often hurt a lot.