Is it safe for me to bike with it or should I wear a regular helmet?
They are designed for different things: climbing helmets have more protection against sharp object penetrating the shell from directly above, while bicycle helmets have more protection for a high force hit with a blunt object (the ground, car bonnets etc) from the sides, front and back.
If helmets are required where you live, unless the climbing helmet meets the standards (EN 1078, CPSC etc) required by your country for cycling helmets, (some might, I don't know) you are not riding legally.
Even the safety benefit of wearing a helmet is hotly debated (refer What are the reasons for and against wearing a bicycle helmet?; wearing the wrong type of helmet is an unproductive debate that will have no useful outcome. Probably what will not be debated too hotly is "If you are in an accident, wearing a climbing helmet is safer than not wearing a helmet" and "It's best to wear a bicycle helmet that meets a recognized standard".
I suggest you refer to discussions in related questions such as are ski helmets safe and Would a military helmet make a safe alternative to a bicycle helmet?.