There is a significant difference in that woke, like SJW, is for the most part a derogatory term. It would be as if the word "christian" was a word created by athiests specifically to refer to those who did not practice what they preached. Any discussion of the merits of "wokeness" is approaching the subject in a biased and unproductive way.

I would suggest that those you would describe as woke have, on average, just as much belief in the importance of mercy and grace as you do, and would likely believe they have more. The difference is who is seen as worthy of that grace and mercy. Is it grace to allow someone who has made unwanted sexual advances in the workplace a second chance? Or is it mercy to those who were victimized to remove the problem? Obviously this is a crude and simplified example, as examples often are, but it illustrates the point.