Being sorry that somebody else felt bad is the beginning of an apology. But it's not the end of one. You can't just say to yourself, "Well now I'll never make somebody feel bad again. Problem solved!"
When you fuck up and hurt someone, people don't want just to hear about you feeling bad. That's making it about your pain instead of theirs. It's asking for people to pat you on the head and tell you it's all ok. That's kinda weak.
They want to hear what you've learned, and why this won't be a problem again.