God says in the Bible, "I, even I, am the one who wipes out your transgressions for My own sake; and I will not remember your sins," (Isaiah 43:25 Heb. 8:12; 10:17). God chooses not to remember our sins. He chooses never to bring them up again. He doesn't forget because He cannot (1 John 3:20). However, He chooses not to remember them. That is, He will never bring them up again. Why? Because they have been forgiven in His Son Jesus (1 John 1:9). To remember is something you do on purpose. To forget is something you do on accident. We can't force ourselves to forget. But, we can certainly force ourselves to remember. So, when God says He will remember our sins no more, He is saying he will never bring them up again. This is why, when you read Hebrews 11, you will see a listing of great people of faith: Abraham, Moses, David, etc. Yet, none of their sins are listed even though they were sinners. Why? Because God remembers their sins no more.
One other point: Jesus told us to forgive as the Father has forgiven us (Eph. 4:32). If someone does something wrong against you, then later asks for forgiveness and you forgive him, you are never supposed to bring up that offense again…ever! Maybe you won't ever forget the offense, but then again you can choose not to remember it. This is what forgiveness means for those of us who have been forgiven by Jesus.
Eph. 4:32, "And be kind to one another, tender-hearted, forgiving each other, just as God in Christ also has forgiven you."
1 John 3:19-20, "We shall know by this that we are of the truth, and shall assure our heart before Him, 20 in whatever our heart condemns us; for God is greater than our heart, and knows all things."