Most people don't realize that there's a subtle difference between saying "I'm sorry." and "I apologize." An apology is a formal admission of doing something that shouldn't have been done. It may or may not be heartfelt i.e. a person may apologize without truly feeling remorse.
On the other hand, saying "I am sorry." is a truer admission of regret. It is what we may call a heartfelt apology. If someone says they are sorry but do not truly feel remorse, then they are said to be lying. However, if someone apologizes but are not truly ruing their actions, they are not lying.
"I'm sorry"' is also used to express sympathy. For example, I'm sorry for your loss. There is no such usage for I apologize. An apology is only for wrongdoing.
edit Examples of the difference between saying "I apologize" and "I'm sorry"
- When Karen's boyfriend cheated on her, she got angry and smashed the windshield of his car. She later apologized. In this example, Karen does not really regret her action because she feels it was justified.
- Politicians often apologize for their gaffes but they seldom say "I am sorry." because saying so makes them appear weak. For example, Joe Barton famously apologized to BP for what he called a $20 billion "government shakedown" of the company. After facing harsh criticism for his remarks, he apologized for his remarks, saying that they had been misconstrued. Jo Bonner of Alabama's 1st congressional district called Barton's apology "half-hearted" and believed Barton was more interested in saving his job than making things right.
- On the other hand, saying "I am sorry" is more apt in situations when you cause another person emotional distress. For example, I am sorry I recorded the soccer match over our wedding video.
- Another obvious scenario where there is a difference is when you are expressing sympathy. e.g. I am sorry your children fell sick before the camping trip. It would be absurd to say "I apologize" in this case because you weren't responsible.
edit Quotes about apologies
- "It is a good rule in life never to apologize. The right sort of people do not want apologies, and the wrong sort take a mean advantage of them." - P.G. Wodehouse
- A stiff apology is a second insult.... The injured party does not want to be compensated because he has been wronged; he wants to be healed because he has been hurt. - G.K. Chesterton
- True remorse is never just a regret over consequence; it is a regret over motive. - Mignon McLaughlin, The Neurotic's Notebook, 1960