No you should not use a bicycle helmet for climbing. They are designed for different types of impacts and will not provide you with proper protection outside of their designed activity.
Bicycle helmets are designed for a SINGLE ground impact. Like modern cars they are designed to crumple and absorb the energy from an impact. They probably provide the best protection of any type of helmet because of this design. Once they've had an impact they are now completely useless though. This is not a big deal after a bike accident as you can probably stop riding the bike and walk or get a ride home.
Climbing helmets on the other hand are designed for multiple impacts. They do not absorb impact energy like a bicycle helmet but they can provide protection even after an impact. This is important when you're climbing because unlike biking you can't always stop climbing just because you've had an accident. You've got to complete your ascent or return to the ground. Additionally, most head injuries from climbing are not from the climber falling but rather from rocks falling on their heads from above.
So imagine you've worn your bike helmet to climb and managed to bang your head on the wall. The bike helmet is crumpled now and will provide no protection now. Suddenly some rocks fall from above. You would be severely injured as the bike helmet would not absorb the impacts or even deflect very well at this point.
You can't really compare the standards as they are for different activities and requirements. As stated above the cycling helmet standards do not require multiple impacts. EN climbing helmet testing is done under EN 12492: 2000 and these tests require multiple impacts to the helmet. Specifically a 5kg hemispherical striker is dropped at 30 and 120 degrees (from the horizontal plane) on the top of the helmet from a height of 2 meters. A flat 5kg striker is then hit on the front, sides, and rear of the helmet. There are also penetration tests done.
The maximum force that can be transferred to the head 10kN. So climbing helmets are not designed to absorb as much of an impact. They are designed for multiple impacts of smaller force instead.
I am not as familiar with bicycling helmet standards, just that their design does not require multiple impacts. If you've ever seen a cycling helmet after a ground impact you can see clearly it would not provide protection for a second impact. Rather, they are designed to absorb much higher levels of force by crumpling. CPSC standards permit only 3kN of force to be transferred to the head. I'm not sure what weight and how they test this though.
So the short version is that a cycling helmet must not transfer more than 3kN of force while a climbing helmet may transfer up to 10kN.
There are many more standards setting bodies for cycling helmets as well. I only used the CPSC because it's the most common one on the market. Snell has a much more stringent standard.
I have seen some of the foam climbing helmets but I'm not sure what the design theory is behind them. I suppose they would absorb more impact from any single impact. Once the foam has been impacted it would not provide that level of absorption again though. The shell of the helmet would still need to meet the multiple impact requirements to be approved by EN.
The other issue is that most cycling helmets have fairly wide ventilation slits. These are far to large for climbing and could easily allow falling rocks to penetrate to the head.
There are now some multi-sport helmets on the market but these are intended for climbing, canyoneering, and kayaking/rafting. Cycling and motoring helmets will always require higher absorption levels because higher speeds (and thus force) are involved.