There are many differences.
Bicycle helmets are thicker and constructed to crumple (like modern cars) due to the higher crash speeds (with a heavier mass (your flailing body)) behind it. They also have a thinner shell to better allow the EPS (foam) to deform and absorb the impact. They also have large vents for airflow. After one crash the foam is permanently deformed and the helmet should be discarded.
Climbing helmets come in a two main varieties. There's the "hard-hat" version which is just a hard shell suspended above your noggin by webbing. It's heavy, but very durable. These are the types you most often see in rental fleets. Often cheaper, but heavier. These work great for things that come flying down at you, but not as good for tumbles since the sides and back aren't well protected. These can take a beating.
Hybrid helmet combine a "hardhat"-style with some foam. Think Petzl Elios, with foam on top and just hardhat on the sides. The shell is thicker than a bicycle helmet, though. These are a blend, and thus come with some of the pros of a bike helmet (lighter, better impact protection) and pros of a hardhat (more durable). But they also have some of the cons (after a significant drop the foam will be deformed and will lose its protective qualities).
So should you use a bike helmet for climbing? Consider the following:
-You'll probably drop or significantly bump your helmet at some point. This, in all liklihood, could make it so that you have made your one helmet inoperable for both climbing and for biking: by trying to save a few bucks you ended up wasting a perfectly good bike helmet whilst climbing
-the vents could allow a pointy rock to pierce your skull and turn you into a crimson fountain
-the side impact of your bike helmet is probably better than a climbing helmet, though!
-How much is your head worth?
Not to pimp my own product (or rather, to pimp my own product), but I have one for sale that I'll sell for $35 (I pay shipping). New. http://www.geartrade.com/item/266530/new-camp-rockstar-helmet-unused