to apologize it's the action, while to be sorry it's the feeling, so you use the second when you want share the fact that you're feeling sorry for something. for example a friend tells you that he had a bad day, and you say you're sorry for him. for the apologize you're actually doing the action of being sorry, something like that, i don't know if it's clear^^

Apologize is a verb (an action verb), so it expresses an action. Sorry is an adjective, so it expresses a state or a feeling, the expression "to be sorry" is a state verb.

Same meaning and usage. I am sorry means that you apologized. However, if you want to split hairs, Onijin is quite correct. Also "sorry" means that you really feel regret. You can apologize and not feel sorry at the same time! For example, politicians never say sorry, they apologize because they have to do it, not because they are truly feeling sorry.

Just to add an information on sorry and apologize If the person is sorry, - this means that the person made the mistake and expresses simpathy on the mistake he made if the person is apologizing - this means that he did not do wrong act but feels sorry for the other person Example: I am sorry i lost the money that you gave me. I apologize that your purse was stolen yesterday.