Well, Entschuldigung means excuse me and es tut mir leid literally means it suffers to me/or sorry in English. So, I would use Entschuldigung if I wanted to say excuse me as in excuse me, you're in my way, or excuse me I want to say something, or excuse me you should be listening when I'm talking (to my students) or excuse me to the waiter/waitress in a restaurant if I want to order something. I would use sorry if apologising for something I'd eone wrong. e.g. sorry I don't understand or sorry for not handing in my homework on time, or as the others on here have said, sorry to hear that Fido's died. A stronger form of sorry in this case, would be es tut mir sehr leid-I'm very sorry or if you want to be VERY polite and formal-ich bedauere mich (I regret)-e.g. Tony Blair would use this expression to the public after the latest debacle to do with Iraq.
Hope this answer helps!
PS. I apologise to the Germans if I'm wrong but this is what my teachers and lecturers (Germans) have taught me.
Me-former German degree student at Goldmsiths College, University of London.