Mercy and grace, in as much as we use these words to refer to God, can refer to the same thing. That is, both mercy and grace--and love and justice and truth and any other word that can rightly be used to refer to God--refer to God as God comes to us. I am tempted to say that they are attributes or characteristics of God, but that would not be quite right. God merely is, or more correctly, God exists as Himself beyond even existence. And yet, God also comes to us and makes Himself known to us.
God does not change. God cannot be divided or enumerated by characteristics--as a scientist might a new species. Most of what we can with any confidence assert about God must be stated apophatically: What God is not. God is immortal (He does not die), God is immutable (He does not change), God is invisible, etc. And yet, God comes to us. The profound mystery of God’s relationship with human beings is that the unknowable God makes himself known, the Invisible reveals Himself.
In this sense, mercy and grace refer to the same thing, in that both words refer to God's coming to us. They refer to what is experienced or known of God. They refer to the works or the manifestation or the life, light and energy of God--all words used to talk about how we experience God's coming to us. And because God’s actions, God’s works, come out of who God is, we can say things like God is merciful (because we experience mercy from God), God is gracious (because we experience grace from God), God is light, God is love, God is just, etc. (because these are what we experience from God). However, all of these attributes are conditioned by human experience; so they in no way limit, and certainly do not exhaust, who God is. Words are of very limited use when speaking about God because words have meaning only in our very limited human experiences. (What does green mean to someone who is blind? Our experience gives meaning to words.)
And yet, mercy and grace may differ, or refer to slightly different experiences of the God who does not differ, does not change. I read somewhere once that grace refers to God’s coming to us despite our unworthiness, lowliness and sin; and mercy refers to God’s coming to us because we are suffering and miserable. Despite and because. When we talk about God’s grace we are emphasizing God’s freedom from obligation and our complete lack of right: God owes us nothing, and we owe God everything. Grace is God’s coming to us in spite of our debt. Grace refers to God’s magnanimity, condescension, and freedom from obligation as he comes to us, comforts us, reveals Himself to us and saves us.
Mercy, in a sense, is a bolder word. Mercy implies compassion. Mercy asks God to see our suffering, our pain, our miserable condition; and mercy asks (here we get very bold) for compassion. We ask the passionless God to share in our suffering; we ask God to show compassion, to act because he shares our suffering, to have mercy. How is this possible? How can the immutable God suffer with us? How can we be so bold as to ask for such a thing?
Mystery: God became man without ceasing to be God. Jesus Christ is the Theanthropos, the God-man. In Christ, God experiences change (without changing), God shares in our suffering (without suffering), God enters our misery (without being miserable). Like I said, it’s a mystery. God comes down without leaving heaven, God suffers as a man without ceasing to be the impassable God.
In Christ, because of Christ, and through Christ we can ask for mercy, we can ask for compassion. In Christ we have boldness to ask God to come to us not only despite, but also because, because God Himself knows and shares our suffering, because God has compassion.
But this brings us back to the beginning. In our human experience, despite and because seem to refer to the same thing. God comes to us despite our sin, and we suffer because of our sin. Because God suffers with us, He comes to us in our suffering, despite the fact that our suffering is caused by sin. When we pray the Jesus Prayer, “Lord Jesus Christ, Son of God, have mercy on me, the sinner,” we are praying for both mercy and grace. We are asking God to be Himself to us, despite our sin and because of our need.