What is the distinction between a mere apology and true biblical repentance?

Rob Ventura explains in this video that a mere apology conveys how a person feels, that they feel bad for what they did. Instead of apologizing, they need to repent. 

This involves explaining that they have sinned against that person and ask for the other person's forgiveness. Repentance involves an understanding that you have sinned against another person. 

James 5:16 (NKJV) - "Confess your trespasses[a] to one another, and pray for one another, that you may be healed. The effective, fervent prayer of a righteous man avails much."

I've seen husbands weep and wives weep. Just as they get the biblical principles and they're confessing sin to one another, I'd stop and say, now John, you need to confess that to Mary. Right there, they express repentance. You need to look at her in the eyes and say, would you please forgive me? Now I apologize, okay.

Apologies, that's not repentance. If you say, I'm sorry, you just told me how you feel. Okay, repentance is, I have sinned against you. Would you, John, tell Mary, you have sinned against her. You haven't been her head, you haven't done what you're supposed to do, and he confesses that to her.

She receives it. We see repentance in action, We see forgiveness in action, and I tell you that's a glorious thing.