I'd also venture to say that we are all still learning. Despite the amount of knowledge we possess, there's more mistakes to be made. If we're really paying attention, great lessons can come from them.

If it's a situation where a person has been wronged, then yes, an apology is in order. But instead of the offender continuing to verbalize regret, sincerity should be demonstrated with actions. Only by showing a change through behavior does the apology become believable, and then the process starts for everyone to get past it.

When I was a kid I used to take piano lessons and I was really serious about it. Yeah, I was one of those kids. Anyway, I remember after every recital where I'd make a mistake I'd tell my parents. They'd say, “Great job honey!” And my response would be “Oh, no, I messed up at suchandsuch part.” Eventually my mom told me to stop mentioning mistakes! She said she never noticed them first of all, and even if she did by the time it was over they were forgotten. All I was doing was re-highlighting a mistake no one cared about.

Deal with that mistake in your head, then forget about it. If something deserves an apology, apologize. Then move on.