Adapted from Rabbi Sari Laufer and Yael Farber, Stephen Wise Temple, 5784

Babylonian Talmud Yoma 87a
Rav was once offended by a certain butcher and when, by the eve of Yom Kippur, the butcher did not come to him, Rav said: I will go to the butcher and reconcile with him. Rabbi Huna, a student and colleague, met Rav along the way and asked: Sir, where are you going so close to Yom Kippur? Rav answered: I am going to reconcile with So-and-So. Rabbi Huna thought: Abba (aka Rav) is about to kill someone. Rav went to the butcher and remained standing in front of him as the butcher was sitting and chopping an animal’s head. The butcher raised his eyes and saw Rav. Then he said: You are Abba. Go away! I will have nothing to do with you. While he was chopping the head, a bone flew off and killed him.

Social psychologists teach us that when it comes to rupture and repair in human relations, there are two types of cultures. There are cultures of forgiveness and cultures of reconciliation.
Cultures of forgiveness are usually cultures with a strong sense of the individual. The offense was unfair and unjust. It creates a debt to the aggrieved. The apology of the offender allows the release of guilt. The apology pays off the debt and heals the insult and the injury. It allows for the aggrieved to achieve emotional closure. Where forgiveness is at a premium, the offender must do the work of accepting blame and committing to a new path. Only by righting the self, can the offender hope to right the relationship. There is clarity in what has transpired which paves the path for a redeemed future. The forgiveness offered by the aggrieved is an act of empathy with the contrition of the apologizing offender.
By contrast, cultures of reconciliation tend to focus on the collective. Who I am is entirely bound up by who we are, the bonds we share, and the common goals to which we aspire. When a relationship is ruptured, a premium is placed on re-achieving harmony with those around me. Here, value is placed on letting go. Introspection by the offender, and offers of apologies are, of course, always welcome. But not all are able or willing to do that work, and those cannot be pre-conditions for restoring harmony. The closer the dependence between offender and aggrieved, the more important it is to forego processes we would call asking/granting forgiveness, and the greater the need for all parties to move to reconciliation. The deep cleansing of introspection and forgiveness may not be achieved.
--Rabbi Dr. Joshua Berman

A 2015 study suggests that there are two types of forgiveness:

• decisional forgiveness: making a conscious decision to let go of hurt feelings, such as anger and resentment, putting them in the past, and moving forward free of the effects those feelings can bring
  
• emotional forgiveness: replacing negative emotions toward the person who has wronged you with positive ones such as sympathy, compassion, or empathy
  

Experts in this study suggest that emotional forgiveness can lead to higher levels of forgetting than decisional forgiveness or no forgiveness.
Dr. Kurt Smith, PsyD
https://psychcentral.com/health/reasons-to-forgive-but-not-forget