"Excuse me" is asking for permission.
"Sorry" is asking for forgiveness.
"Sorry" expresses more regret than excuse me does.
Examples:
- If I accidentally step on your toe while I'm trying to get to the door, I'll say: "I'm sorry!"
- If one of your relatives or friends died, I will probably say: "I'm sorry for your loss."
- If I'm trying to leave the room and you're in my way, I'll say "Excuse me."
- If I want you to show me the direction or guide me, I'll say "Excuse me.