I haven't read NVC, but I am peripherally familiar. I would say that the two are only related so far as they are books about communication; for people who only read sci-fi, this might seem to be apples and apples... NVC is prescriptive while DC is about developing a greater understanding of intent and the motivations behind our words.
I was about to summarize, but instead, I'm going to copy and paste from the web:
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Difficult conversations are anything that someone does not want to talk about, such as asking for a raise or complaining to a neighbor about his barking dog. People are usually reluctant to open a difficult conversation out of fear of the consequences. Typically, when the conversation does occur the parties think and feel a lot more than they actually say.
Underlying every difficult conversation are actually three deeper conversations. The "What happened?" conversation usually involves disagreement over what happened, what should happen, and who is to blame. The feelings conversation is about the parties' emotions, and their validity. The identity conversation is an internal conversation that each party has with herself, over what the situation tells her about who she is. The authors identify common errors that people make in these sorts of conversations. The key to having effective, productive conversations is to recognize the presence of these deeper conversations, avoid the common errors, and turn difficult conversations into learning conversations.
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My favourite example is a scenario in which you have been consistently working late and missing dinner - and your partner is pissed. You feel tired and angry that your hard work is unappreciated and that they are just adding to your frustrations. This is roughly where most couples would fight.
The book helps you understand that when your partner is angry that you've missed dinner... well, unless you've been wasting away of a nutritional disease, they aren't concerned that you're hungry. They feel lonely and disrespected. They are aware that in the early days of a relationship, all of the demons in hell couldn't have kept you away... so what's changed? They are worried for the future of your relationship. They are worried that they've been replaced, physically or emotionally.
And part of this is the questions you have to ask yourself; is your partner correct to be concerned? Has something intangible changed? Am I hiding and using work as a cover?
After reading the book, I had powerful new tools to understand and correct the problem. Which is a big deal because these sorts of problems happen all of the freaking time. And if I can make even some of the problems in my life resolve faster with a better outcome, then I'm truly winning.
Honestly, the book was so powerful for me that when I finished, my primary emotion was anger at myself. I suddenly had to confront the idea that most of the major problems and conflicts in my life had been far more my fault than I'd ever considered... and that if I'd done some simple things differently eg. thought about what someone was really saying, there are people gone from my life that might have otherwise still been there today.