Somewhere in his diaries, Kierkegaard thinks forgiveness by an omniscient being is paradoxical. (Of course, that's no reason to deny either divine forgiveness or divine omniscience, for Kierkegaard—paradox here is good.) One way of filling out the paradox is that forgiveness is a kind of forgetting, and an omniscient being can't do that. There is a story, quite possibly anecdotal, of a girl who claimed to have visions of Jesus. Eventually she has an audience with her bishop, who to check if the visions were genuine, asked her to ask Jesus what he (the bishop) confessed in his last confession. Next week the girl comes back: "Jesus said he forgot." And of course our language has the phrase: "Forgive and forget."
However, forgiveness is actually very different from forgetting. First of all, forgetting is insufficient for forgiveness. Forgiveness (when valid) has a crucial normative consequence—there is something that the forgiven malefactor no longer owes her forgiving victim. Forgetting has no such normative consequence. I may be glad that my misdeed has been covered over by mist in your memory, but I am in no way off the hook. If anything, I am in a tougher place once you have forgotten, because the only way I can get forgiveness from you is by reminding you of what I have done, and that might be undesirable (it might cause pain to you again).
Moreover, forgetting has no place in ideal cases of forgiveness. For if you have forgetten the ill I have done you, you surely have likewise forgetten that you have forgiven me that ill, unless your logical skills have gone haywire. But now it is possible that you will come across evidence of what I did to you, without at the same time coming upon any evidence of your forgiveness. And if you do not know that you have forgiven me this offense, you may well experience the kind of resentment towards me that you had put away when you forgave me, and even seek vengeance. Forgiveness includes an objective and a subjective component: the objective component is a certain normative canceling of debt (as it were?), while the subjective is a putting away of resentment and a surrender of the desire for vengeance. But if you do not remember having forgiven, the resentment may return, until you go through the subjective component of forgiveness again.
In the ideal case, when you have forgiven me, you keep track of your new commitment not to resent such-and-such a deed and of my new normative status as "forgiven former malefactor".
This, however, carries a danger with it. For if you keep track of what you have forgiven me, then surely you run the danger of your priding yourself on your moral superiority in not holding me bound. To dwell on having forgiven is dangerous—forgiveness does, after all, put us in a position of superiority, and there is a danger that our forgiveness will be haughty and humiliating to the one forgiven. The cure for this is love, and we sinners have an additional help, which is meditation on our own sinfulness—we do not forgive haughtily, but we forgive as we wish to be forgiven.