There is a subtle difference between saying "I'm sorry" and "I apologize". An apology is a formal admission of a wrongdoing. It may or may not be heartfelt — i.e., a person may apologize without feeling remorseful. On the other hand, saying "I am sorry" is usually seen as being a truer admission of regret. It is what is called a "heartfelt apology." If someone says he is sorry but does not feel any remorse, then he is said to be lying.

"I'm sorry" is also used to express sympathy. For example, "I'm sorry for your loss" can communicate sympathy following the death of a loved one. There is no such usage for "I apologize." An apology is only for wrongdoing.