The dictionary definitions seem semantically identical. Do you use one instead of the other in certain situations?
(I am a native English speaker who is learning German)
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"Schuld" means debt/fault/blame. if you are in debt / to blame ("schuldig") you have to get ... basically get out of it. so "ent-schuld-igen" is getting out of it. If you hit someone, you are to blame and you can't get "entschuldigen" yourself, you have to ask for "Entschuldigung" (Noun, "um Entschuldigung bitten"). To some very very sensitive people it may appear rude to say "Entschuldigung." ("Sorry") to unblame yourself, instead of asking for it to the one who blames you. Most people don't even notice the difference. In daily life, it doesn't really matter.
"Ich entschuldige mich." -> Sorry, "unblame" yourself (Deutsche Bahn uses this...)
"Ich bitte um Entschuldigung." -> ask to get "unblamed" (generally considered more formal)