as you apologise someone, that means that you have done s.th wrong, and made him angry or upset, or may be disappionted of yourself.
but when you make an excuse, it means that s.b wants you to do him a favor, and you don't want to.
Yes, just saying what matilda did but in another way. An apology is admitting that you did something wrong and want forgiveness, where as an excuse is acknowledging that something wrong was done, but that it wasn't your fault. For example:
Apology: "I realise that I should have completed my work for today. I am sorry and I will try harder in future."
Excuse: "I would have had my work completed for today, but I had other things to do which couldn't be ignored."
Also, as matilda said, an excuse can be used to avoid doing something you don't want to do, such as tidying your bedroom. Generally, an apology is something which is genuine, where as excuses tend to be lies to avoid unwanted situations.