The biggest difference I see about being under law and being under grace is same as difference between fear and love.
The message of law is condemnation but the message of grace is love.
1 John 4:7-21 talks about God's love (agape love or unconditional love)
18There is no fear in love; but perfect love casts out fear, because fear involves punishment, and the one who fears is not perfected in love.
When Christ gives us freedom from law, we step into love from fear. "fear involves punishment" - law says if I break it I will be punished. Grace says God loves and accepts us unconditionally, regardless of how we behave!
Does this mean that we can live any way we want? NO!
19We love, because He first loved us.
When we realize the love of God, we will love Him back. He proved His love by dying on the cross. We prove our love by walking according to the Spirit who dwells in us, by abiding in Him, by allowing Him to live His life through us.
Romans 8:15 says, For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, "Abba! Father!"
Verse 16: The Spirit Himself testifies with our spirit that we are children of God.
Verse17: and if children, heirs also, heirs of God and fellow heirs with Christ, if indeed we suffer with Him so that we may also be glorified with Him.
We are adopted as sons and daughters.
A slave serves his master out of fear; a child of God serves God (Father) out of love.
Therefore let us draw near with confidence to the throne of grace, so that we may receive mercy and find grace to help in time of need. (Hebrews 4:16)
Therefore you are no longer a slave, but a son; and if a son, then an heir through God. (Galatians 4:7)
Jesus said "Love your neighbor as yourself". This is an impossibility with law. But this is a possibility when God's love being expressed through us.