One example of a moral truth, used in philosophical ethics, is the statement Raping children for fun is wrong.
Christians would say Raping children for fun is sin. The concept of sin implies a Divine Command Giver to whom persons are responsible and accountable.
Imagine a person, whom we'll call 'X', gets caught while raping a child for their own pleasure. They may admit that they did this. From their admittance it does not necessarily follow that they believe what they did was wrong. Indeed, if they are an intellectual atheist, or even a village atheist (see Nietzsche), they may not believe morality exists at all (viz., objective morality),
To admit something it just to say, "Yes, I did this."
If someone confesses something, they have gone beyond mere admittance. If someone confesses "I raped a child for fun," there can be guilt involved. The person who merely admits they did something is often found out. The person who confesses something is often moved by guilt, and comes forth. An admittance is often involuntary. A confession is often voluntary.
If someone confesses a sinful action, how can we tell if their confession is genuine? We can discern genuineness when it is accompanied by remorse. By sadness. Even, grief. About not only how our sin may have hurt others, but how it has disappointed God, and created a breach in the God-relationship.
Psalm 51 exemplifies this, where David is writing from the darkness of his sexual and murderous sin. Ps. 51 is confessional, remorseful, and repentant.
Remorse usually leads to repentance. When remorseful people confess their sin to others, and to God, they run after "doing a 180." They get the help they need to morally and spiritually move in the other direction. The seeds sown in the ongoing acts of confessional remorse begin to emerge and grow from the soil of genuine repentance.