I don't recall Jesus ever saying to someone, "Don't worry about your sins. It's not really a big deal. Everybody does it." And I don't remember him saying, "Nobody's perfect" when speaking to someone who was conscious of sin. (Though he did challenge, "Who among you is without sin?" when speaking to people who were conscious only of someone else's sins.) With Jesus, forgiveness did not come with excuses or a denial that a thing was significantly wrong; it came with redemption. "Go and sin no more."
And Jesus did not forgive so that the memory of being wronged would stop bothering him -- that is, for reasons that had to do with himself. The memory of being wronged may bother us, and may send us looking for how to get rid of the memory of being wronged. But simply forgetting the wrong is one of those so-called shortcuts that doesn't really work, at least not for a wrong that is big enough to make a difference in our lives, a wrong that insists on being noticed.
Forgiveness works together with love. It is firm in its stand that the wrong should stop. It insists that the damage should be repaired as much as possible -- by the person who did the wrong. It asks some positive action or change from the offender -- something to begin to restore the soul not just of the offended but also of the offender.
Forgiveness works to restore the love and trust that were broken. The call to forgiveness is a call to repentance: it calls on the one who broke that bond to take responsibility for himself and admit wrong instead of fleeing to excuses and the blame of others. So those kinds of "forgiveness" where the wrong is excused, and the wrongdoer's actions are explained away, is more of a justification for evil, something that does not deserve the name of forgiveness. That justification of evil only serves to encourage the person in the habit of wrong and to strengthen his judgment that he is blameless for his own actions.
As for forgetting the wrongs that were done us: if the bonds of love and trust are not restored, and the wrong is not called on to stop, and the person is still around but without love or trust -- or redemption -- then it is an incomplete forgiveness, an imperfect forgiveness that has not obtained its goal.