Is there a difference between calming my mind and forgiving?
Yes, there is a difference between the two. Calming the mind is not a moral virtue. Forgiving is a moral virtue, which means that the focus of forgiving actually is on the one who offended, not on the self. As you forgive, you begin to think about this other person in new ways, to feel softer feelings toward this person, and to behave in a way that is civil and not hurtful. When you calm your mind you could be ignoring the other, putting the other out of your mind. Of course, this does not always happen when you calm your mind, but it can happen. Thus, calming the mind does not necessarily lend itself to a focus on the other, and in a positive way, as forgiveness does.
For additional information, see What is Forgiveness?