I apologise is generally more formal and I'm sorry more informal.

I think in some cases they have the same meaning but "sorry" has more meanings than " apologize". "I Apologize" mean you feel sorry for somebody, some issue when it's you who make the problem happens or ruin something. and"I sorry" :keenly sorry or regretful; feeling or expressing sorrow or pity, having regret or sorrow or a sense of loss over something done or undone,..

"I'm sorry" is how you feel. "I apologize" is what you do. Functionally, you can use either in the same situation, but "I apologize" is clearly formal and credible. You would use "I'm sorry" in daily speech.

I'm sorry(more general) I'm sorry for being late / I'm sorry for the lost of your dog / I'm sorry for speaking too quickly I apologize(feel regret) (seeking forgiveness)