A lot can be said about the difference between law and grace but one revelation that has changed practically all aspects of my life is found on the sermon of the mount.
Law attempts to control and change behavior while grace deals with the heart. The law says “do not commit adultery” but when grace comes, grace is not concerned about what you have done or didn’t do; grace is concerned about the heart struggling with lust for other people’s wife (or husband).
While most people will live their entire life without killing someone, grace is concerned about a heart that carries the heaviness and burden caused by hatred, bitterness and resentment.
One under the law will rejoice and celebrate the victory over a temptation; while the one under grace will come to God in humbleness and ask Him to reveal the condition of his or her heart as to why the temptation was there in the first place.
The one under the law will have to fight the same temptation on an ongoing basis and require a safe environment and sufficient protection to win. The one under grace will experience deep inner healing and will see continuous victory as a healed heart is not a good environment for temptation.
One under the law is prone to look down and judge others while the one under grace will have compassion as he or she understands it is not a behavior problem but a heart problem.
Law tries to control and changes while grace heals and sets free and allows the change to flow as a result/fruit. Maybe this is why the WORK of the flesh is contrasted with the FRUIT not work of the spirit in Galatians 5.
Take this furtherer and see how many argument will cease by learning to listen to the heart instead of fighting over words being used.
A lot of teachings make law and grace such a difficult concept but as soon as we translate it to behavior and heart, it starts to make sense. After all God said He looks at the heart while people look at the outward appearance. We can almost say that living by grace is seeing life and people the way God does.