Actually, "Es tut mir Leid" and also "Entschuldigung" are both replacing the old "Verzeihen sie mir / Verzeihung". "Verzeihung" is/was strictly used to apologize for your own actions. "Es tut mir Leid" is in fact also used to apologize for your own actions, however, the cause for the apology is then more serious (it comes close to the more formal "Ich entschuldige mich" then). "Entschuldigung" (or also "T´schuldigung" in modern slang) is appropriate in the daily routine.

Actually, "Es tut mir leid" and also "Entschuldigung" are both replacing the old "Verzeihen Sie mir / Verzeihung". "Verzeihung" is/was strictly used to apologize for your own actions. "Es tut mir leid" is in fact also used to apologize for your own actions, however, the cause for the apology is then more serious (it comes close to the more formal "Ich entschuldige mich" then). "Entschuldigung" (or also "’tschuldigung" in modern slang) is appropriate in the daily routine.