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)
"Entschuldigen" is the verb form.
"Das Entschuldigen" is an example of a gerund (Gerundium), which is simply adding the neuter article "das" to a verb and then capitalizing it to make the verb act as a noun. In this case, "das Entschuldigen" is the act of excusing, whereas "die Entschuldigung" is the excuse itself.
"Das Entschuldigen" could go on and on while someone continues asking for "die Entschuldigung".
"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)