Where did you dig that one up? Actually though, I believe those words do have different meanings. Well they do to me at any rate.
I use them differently, 'I apologize' to accept responsibility for something I did poorly & I use 'I'm sorry' to express compassion. But mart, sometimes people say to me "it wasn't your fault!" when I say sorry, so maybe your usage is more common.
To me they're the same, but at a funeral would say, sorry for your loss.but not apologize. Can say sorry for forgetting your birthday or I apologize but sorry is better.