Few people understand why I get angry after they say "I'm sorry." After all, they just apologized, right? Wrong. "I'm sorry" tells me how they feel. Which may or may not make me feel better, but has no bearing on the situation. "I apologize" is an active verb, signifying that they are actually doing something, in effort to make amends.
And yes, I am a hypocrite sometimes. I say "I'm sorry" when I intend something more meaningful. But when she's there crying, or my father looks at the crashed car, or (as ymelup points out) something is beyond a doubt my fault, I expound on this and explain that I have more regret than to merely point out I feel poorly about the subject. It carries a higher note of honor, in my opinion, when you are up front about misdoings and if you express a true desire to change the situation.