Entschuldigung means in English "apology", and it may mean "excuse me" as a start to a question (Entschuldigung, ich habe eine Frage.)
Tut mir leid means in English "I am sorry". This may overlap with Entschuldigung, but you may also use "tut mir leid" to say, that you feel sorry about sth. or for someone:
Peter ist krank.
Das tut mir sehr leid.