A stand-alone apology holds little power. Though some power exists inside the acknowledgement of wrong-doing, it is what comes after the apology that is most powerful.
Changed behavior is the true apology.
I broke one of my own rules last night, and I immediately apologized once I realized I had done so. I know all of the reasons that built up into making that decision, and I know how to avoid the same mistake in the future.
But I also had to apologize to myself last night, for a different but related part of the circumstance I was in. And find the grace inside my own mind to offer up understanding and to gently remind myself of the reasons why I might make the assumption that I did.
Behavior tells you everything you need to know.