I haven't actually read Getting to Yes, so I could be wrong, but from what I gather (ie what I've heard about it), it very much focuses on persuasion through logic, facts, etc assuming that both parties are rational and a rational argument will persuade, while Never Split the Difference states that this doesn't work because humans are inherently irrational and react emotionally, so you need to be in tune with your counter-parties emotional state (which often simply comes down to that they want to feel like their concerns are being listened to and understood).

You don't want the typical sales pitch that you might hear from a telemarketer or "slick sales person" -- you come away from those drained and annoyed -- instead you want to be listened to and understood.

Never Split the Difference spends some time talking about why more traditional sales practices don't work very well, hence re-education. You want to break the habits that are (apparently) thought in Getting to Yes and instead work on you listening skills, empathy and techniques for learning about your counter-parties emotional state, fears and desires. There's a chapter called something like "getting to no" because once the person has told you no, you can work with them to find out what they actually want and if you can give it to them. Its also very clear that you shouldn't be doing much of the talking, you should let the other person talk while you listen and poke them with tactical statements and questions to get them to focus in on the important details. This is rather different from more traditional sales "pitches" where you talk at them most of the time.