To forbear does not simply mean that we put up with the person who has offended us. We do that also, but in forbearing we go the further step of not taking advantage of our situation. It would be easy to forbear with one another if we were doing so only while looking for the opportunity to exact revenge upon the person. But again, real forgiveness does not think this way. Forgiveness implies a sincerity and comprehensiveness that puts away all anger, wrath, and desire for revenge.
Forgiveness offers acquittal from the offended party to the offending party.
Forgiveness does good in return. Love your enemies, do good to them which hate you. Of course, this is the example given to us by Christ. Christ is the arch-type example in the forgiveness profile. Christ offers forgiveness to us when we have done Him, His Church, and His name great harm.