I'm sorry but I refuse to apologize. sorry = adjective apologize = verb

Well, to apologize is to say sorry. But i'm sorry has another meaning which is used to console someone who has a relative or friends or family member just passed away. However like what chris has said, perhaps apologize sounds more formal than sorry. But generally it means the same.

as far as I'm concerned they are the same thing. perhaps I apologize is a bit more formal

In general usage, I'm sorry is more personal. For instance, you have an employee at work and they have been rude to another employee. You ask them to apologize. They won't say, I apologize unless they are being a smart aleck. They will say, "I'm sorry."

'I apologise' is stronger than 'I'm sorry'.It expresses a deeper level of regret or remorse for something you've done or said.'I'm sorry' is also used to express sympathy to the bereaved.