Bruce C. answered 07/15/20
Retired Technical Writer
I think Westley (above) nailed it.
There are 2 situations where "excuse me" is appropriate. The first is when you incidentally encounter or inconvenience someone else (think bumping into them or accidentally stepping in their way) and you don't really believe that you did anything wrong. The second is when you intentionally interrupt someone or intrude on them for a benign purpose (think - "excuse me but I think that is your car being towed" or "Excuse me, but are you 'John'?"). What characterizes both of these types of situations is that they are benign - you are innocent or acting in good faith.
"Forgive me" can be used in those same situations, but most people reserve the use of this phrase for situations where they are NOT innocent, or know they are behaving badly, or something bad happened. For example, "forgive me for saying so, but that shirt looks terrible on you." Or, "I ate all of the ice cream - I hope you will forgive me." In these situations, you are possibly being rude (#1) or know you did something rude (#2). As the seriousness of the situation escalates, people are more inclined to use "forgive me" rather than "excuse me."
Westley Bud E."excuse me" is a polite method, and or introduction to an intrusion on ones behalf. "forgive me" insinuates that the individual asking forgiveness has done something that he/she requires atonement for.