In the Bible the words love and grace are perhaps used interchangeably as in 2Cor. 13:14 ‘May the grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit be with you all.’ Here we get the sense of the all-encompassing presence of God.
BUT in Biblical Theology grace and love are different. Here’s an example. I love my children very much and want the good for them in their lives. When they were little they wanted to learn how to ride their bikes without the training wheels. Now, I could express my love by saying to them, ‘Oh, I want very much for you to be able to ride your bike. And here, here is your new bike. I wish you well. Go for it.’ That’s love. On the other hand my best move would be to assist them in this training exercise. And so I did. I got them on their bikes, and held the bike for them and went even further by running along side of them as they got the hang of it. And there were times when I let them go that they fell. I was right there to pick them up, dry a couple of tears, maybe apply a band aid and sent them on their way again until they said to me, ‘You can go, dad.’ That is the meaning of grace; love in action, the influence of love, the direction and empowering of love.
In Biblical Theology the grace of God is God’s action. The trinity of God doesn’t need to express grace among themselves. Grace is for those who have fallen from God and who don’t have the ability to get up and get back to God. Grace is God’s moving in their lives to give them that ability.
In the example of the bike lesson my children, by their very connection to me, somehow merited my helping them. But with God, humanity was dead to God until God, by God’s own love, decided to do something to help the creation and us. See, the Bible tells us that while we were sinners and helpless Christ died for us to fulfill the covenant promise of God to bless the whole creation.
I hope this helps to distinguish between love and grace because for a long time I have misused the terms and thus loss much of the meaning of how God in God’s loving character relates to God’s world.
Finally, take a look at this verse from Paul’s letter to the Ephesians (good to read the whole chapter for the best context).
4 But God, who is rich in mercy, out of the great love with which he loved us 5 even when we were dead through our trespasses, made us alive together with Christ —by grace you have been saved— 6 and raised us up with him and seated us with him in the heavenly places in Christ Jesus, 7 so that in the ages to come he might show the immeasurable riches of his grace in kindness toward us in Christ Jesus. 8 For by grace you have been saved through faith, and this is not your own doing; it is the gift of God— 9 not the result of works, so that no one may boast. (NRSV)-Bold and underlines are mine.