It occurred to me several years ago that usage of the word "nope" is much more negative than a simple "no".
"Nope" indicates some aggression. With the pop of the lips at the end, I think it has a note of harshness and seems very dismissive. "Nope". There is even a little puff of air when the word is said. I saw one comment when I was reading up on "no" versus "nope" that said this: "Think of how the lips must close together and this gives nope its terminal feeling in a physical way. Pursed lips. Can't pry them open. Not going to change." It does have a very final connotation.
It also adds a sense of no consideration or thought going into the response. Typically, a "no" can come either right away or after a pause. However, "nope" almost always comes immediately. You wouldn't pause to give consideration to your answer, only to say "nope".
Of course, this is not 100% of the time. Notice it the next time, though. When you hear a "nope", stop and think whether or not the feel of the response would have been the same with a "no". Let me know what you find.