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From what I’ve learned, there is a difference when used as an actual apology.
“I’m sorry” (on its own, without being followed by “you feel that way”) is genuine and tells that the speaker is honestly so.
“I apologize” is formal and kind of fake. They’re likely more sorry they were called out rather than sorry for what they were called out for.
I’ve always thought of it as being the opposite. I only ever use ‘I apologize’ or ‘my apologies’ when I’m sorry about something I did, when it’s my fault. I use them because it feels more sincere to me, and more effectively communicates my intention.
This is a Demetri Martin joke