Expressing ones regret over offending others (or mistakes or failures which impact others) for which you accept accountability (i.e., an apology) typically is addressed to those who were negatively impacted.

Although on the surface apologizing appears to be for the benefits of those that were harmed, in reality, it is often more to assist the apologizer themselves with coming to terms with themselves.

Seeking Resolution

An error in action which causes others to think less of you also causes you to think less of yourself.  The resolution needs to include resolving your own internal bad feelings as much or even perhaps more so.

More directly stated, apologies can often be more important for the person making them than the person receiving it.  When I am crafting my apology one objective is to receive recognition and sympathy.

I not only want to mend my relationship with those that we injured by my missteps but also want to feel better about myself.  If the situation is accepted by the relevant stakeholders there is less for me beat myself up about.

Even Praiseworthy

Acknowledging your error and following on your sword clearly and promptly, including what you are doing about the issue (or have already done) to make things whole again is often viewed as a very positive behavior.

If the choice is being defensive or being open and upfront, requesting forgiveness I always choose the latter.  This should not be viewed as a cynical perspective but rather just a practical one.