Entschuldigung is an apology, an expression of regret as in : Entschuldigung , ich sollte das nicht tun . It could be an excuse ,a reason or explanation put forward to defend or justify a fault or offense as in : Entschuldigung , das habe ich aber nicht so gemeint . An exclamation in the sense of 'I beg your pardon "as for example in the form of a request to a speaker to repeat something because one did not hear or understand it as in : Entschuldigung , ich habe dich akustisch nicht verstanden. Tut mir leid or es tut mir leid this is more a form of expressing sympathy with someone ." I am sorry " kind of equivalent . Es tut mir leid, er war echt ein netter Kerl . Otherwise when the meaning is comparable with 'Entschuldigung' , it will be more of an excuse or a reason or explanation put forward to defend a fault or offense : Es tut mir leid , aber ich schaffe es nicht bis morgen.

I think that's something that can vary locally depending on where you are but the way I see it is that they're both almost the same in general usage. Both imply to some degree that you did something wrong and want to apologize for that. As an example, a kid who's being scolded by their mom for breaking a vase might answer with "Entschuldigung [for having done that]". "Tut mir Leid" works in that senario as well and it's just preference in my opinion. As an example on what not to do: You would not go on a funeral and use "Entschuldigung" to comfort someone since that kind of implies you had a hand in killing the person they're mourning. So "Entschuldigung" mostly is "Sorry [for having done something]". "Tut mir Leid" and especially "Es tut mir Leid" sound about the same stand-alone but can become neutral if you add a phrase to it. I could see someone saying "Tut mir Leid [that your girlfriend dumped you. Want to go get some drinks?]" for example. If you try and say that with "Entschulding" + phrase the implication of it being your fault doesn't go away making it sound like it was your fault that his girlfriend dumped him. That being said you can use "Entschuldigung" as a stand-alone "excuse me" as in, if you want to get someones attention. Just as in "excuse me [random stranger on the street], do you know which way XYZ is?". You can't use "Tut mir Leid for that.

If you want to ask something e.g. in a shop or in the street, you would say: "Entschuldigung, wo ist....." (Excuse me, where is....). If you have hurt somebody you could use both. "Tut mir leid" is like "I am sorry".