You're a child. It is your place to show your mom respect by remaining quiet until she is done speaking. Your mother has authority to interrupt you, on the other hand. It's basic education.
It shows humility to accept correction even if you know it may not be your fault. You don't have to be right all the time. It shows self-control and maturity to apologize and commit to doing better, without admitting to any wrongdoing you did not do.
To apologize is not to admit fault, nor is it weakness. It is to express your sadness for the sorrow another is experiencing.
