Interesting question. I suppose the answer would be different for everyone. For me, I think apologies are important because they tell you what a person is really about. Here's what I mean. We all make mistakes. No matter how hard we try, we will hurt the people we love. And they will hurt us. That's just a fact of life. However, some people are also just jerks. And do you really want those people in your life? How can you tell the difference between someone who really cares about you and made a mistake versus someone who just treats people poorly because that's how they are? The answer, for me, lies in apology.
Here's an example. Years ago, my SIL hurt me very deeply. And she knew it. But she refused to apologize for it or accept any responsibility. She expected me to get over it and just move on. The problem was, I couldn't do that because I believed the reason she didn't apologize was because her actions were intentionally meant to hurt me. Afterall, when someone hurts you then tells you they're not sorry, what else are you supposed to think? And the hurt was so egregious, it made me fear any kind of relationship with her. If her actions were intentional, what would stop her from doing it again? When a person apologizes, you get a good sense that they didn't mean to do it, feel badly and will try very hard not to hurt you again in the future. Her refusal to apologize made me believe she would do it again and wouldn't really care about it. So today, we don't speak. And it's ashame. On the flipside, one of my best friends really hurt my feelings a few months ago. And she knew it. Do you know what she did? She drove to my house unexpectedly after work one day just to see me, hug me, and apologize in person. Now THAT is a person who truly cares.
I try to employ the same rule myself. When I hurt somebody, my husband, my children, my family, my friends - I apologize. I want them to know I care about them and would never do anything to hurt them intentionally. You can't just assume people know that. You have to TELL them how you feel. Life's too short not to.