Dr. Michael J. Hurd, a "psychotherapist, life coach and author" has published some thoughts on the psychological benefits of apologies for both the apologizer and the apologizee. He makes some interesting points, linking apology to a sense of justice:

Children, like adults, need to operate on a sense of justice. "Justice" means sticking to, and honoring, the facts. "Was I wrong in what I said or did? Do the facts prove I'm wrong? If so, I should acknowledge it." An apology is a form of acknowledgment. In essence, when you apologize, you're saying: "I know what I did was wrong, and I regret it."

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The goal isn't to be—or not be—sorry. The goal is to acknowledge the truth and apologize only for what you see as your error. ("Treat yourself to an apology" delmarvanow.com)

He makes other good points, some noted before at Apology Index, about the importance of sincerity, the need to back the words of an apology with action, and the fact that the a true apology benefits the apologizer most of all. Also, an apology "doesn't necessarily have to aim for (or result in) forgiveness."

Worth a read by any student of apology. Check it out.