The reason is that there are lots of people on the edge of the cliff fiddling around with carabiners and other bits of gear and it's quite easy to drop one by accident. Being hit by a lump of metal is pretty unpleasant. Climbing helmets are designed to divert glancing blows from falling objects, not save you from falling.

Often with sports climbing you'll see the person belaying (at the bottom holding the rope) wearing a helmet, while the person half way up the climb isn't - the risk is the climber will drop something on the person at the bottom.