This is going to sound mind-numbingly obvious, but bear with me - repenting to a higher power only makes rational sense if you believe there is a higher power concerned with morality.
In fact, repenting to a higher power only makes sense if you believe that your actions do not only harm other people, but harm God. And God forgiving people for what they've done only makes sense if God is the one really hurt by it. I don't know about you, but I generally don't go around forgiving Bobby for stepping on Susie's toe. It's not my toe, after all. Nothing can give God the right to forgive people unless God is the injured party in every wrongdoing. So the whole fabric of religious repentance only holds together if you believe that God is good and is intimately involved with the sufferings of the world.
Emotionally, repenting to a higher power makes a great deal of sense. Sometimes we can't face the people we've hurt to ask for their forgiveness... sometimes they can't give it. So we can either learn to ignore guilt (which can be surprisingly easy to do - one merely avoids thought), face up to it (not so easy), or seek absolution (which still involves facing up to it).
At any rate, all repentance has to involve taking responsibility for what you've done. It's just a question of how you try to live with yourself afterwards.