A bicycle helmet is not well suited for climbing protection for several reasons. The extent of coverage (as already mentioned) is less than a climbing helmet, the number of ventilation holes increases the possibility of falling rock (or plant material poking through) actually getting to the head, and the materials they are made of make them a "single-use" item.
There are two basic designs for climbing helmets. One is similar to a bike helmet, with a thin plastic cover, styrofoam lining, and a snug fit (usually more stylish and less dorky looking), and the other is similar to a construction helmet, with a hard outer shell and webbing suspension (not as stylish, though they are getting better). The bicycle-style helmet is designed to absorb impact by splitting/shattering. It is designed for a single incident/impact. Once is has dispelled any sort of blow, it is discarded. This poses a problem on a long climb if you hit it early in the trip. You have to throw it away - so what do you do for the rest of the climb? The construction-style helmet is designed to deflect and redirect the force of impact away from your head, and to last for multiple impacts. Consider a falling rock striking each of these helmets. The bicycle-style helmet will absorb the impact of the rock, by shattering, and the rock will continue on its path into your head, though a bit slower now. The construction-style helmet will deflect the rock and it will bounce to the side, hopefully missing your shoulder/ back/etc on its way. If you are concerned about slipping and hitting your head while top-roping, a bicycle-style climbing helmet would work fine. If you are lead-climbing or worried about falling rock, the construction-style helmet is the only safe option.
Climb safely