This is a Demetri Martin joke
LPT: When at a loved ones funeral and someone says “I’m sorry” do not say “apology accepted”.
“I’m sorry” conveys that the situation makes you sad. Usually in an “I’m sad this has happened” kind of way (from the root word 'sorrow")
“I apologize” means you regret your role in the situation.
They’re always quite different. You can say “I’m sorry” for any situation that’s upsetting to you or the other person. And that’s not just because I’m Canadian!
You can only say I apologize about the part you played.
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.
Yeah, that’s my take. “I’m sorry” is about how you feel. It’s like the OP post: you’re expressing sympathy, or remorse - an emotion. If it’s because of something you did, it does double duty as an apology. “I apologize” gives no indication that you’re sorry about what you did, or concede that it was wrong. It’s merely a recognition that harm was done, and giving a concession to the other person to achieve harmony.