Mercy is what grace offers. The grace of God comes to us in our poor sinful condition and offers us the mercy of God when we deserve His wrath. So grace comes to us giving us mercy. I admit it is somewhat semantical, and the terms are virtually interchangeable. There is a subtle difference in the two words.
Mercy has meaning more synonymous with compassion.
It denotes, not simply love, but the love of a sovereign, transcendently superior, one that may do what he will, that may wholly choose whether he will love or no. Now God, who is an infinite Sovereign, who might have chosen whether ever He would love us or no, for Him to love us, this is grace. Grace is God's favour that gives us what we do not deserve.