"Excuse me" is asking for permission. "Sorry" is asking for forgiveness. You say "excuse me" before doing something that might inconvenience someone. You say "sorry" after having inconvenienced someone. Please forgive me is a more formal way of saying "Sorry"

If you step on somebody's toe, or accidentally bump into someone, you say "I'm sorry"! If you hurt someone's feelings you really owe them an apology so you should say "I'm sorry I hurt your feelings". "I'm sorry I said that about you. Will you please forgive me?" If you sneeze or cough, you should say "excuse me". If you reach across the table to get the salt, and you have reached across them (into their space), say "excuse me". If you have made someone angry, and you want everything to be alright between you, you might need to say "Please forgive me, I feel awful that I made you angry, and I want you to forgive me". They will probably say, "I forgive you", or "Oh, it's okay, I'm not angry...I was just a little hurt".

Excuse is more strong word than sorry and this word say that you have a some guilty.