"Stay up" just means that one has stayed awake. "Stay up late" says the general time that one has stayed up until - late. Ex. "It's 2:00pm, and, even though I want to take a nap, I have to stay up to babysit." There, I'm staying up, even though I'm not staying up late. "I have to stay up late to babysit" means I'm staying up late at night, not 2pm or the like. In a lot of cases, people use them interchangeably, but that's the actual difference.

These are basically the same thing in my opinion. You can add the word "late" into the sentence to emphasize... I have to stay up to study. I have to stay up late to study. Same thing, but the second is a little more direct or clearer I guess.

para mi es solamente una cuestion de grado I usually stay up till 23:00. My wife often stays up till 1:00. Sometimes I stay up late and go to sleep with her at 1:00. However I can easily imagine that my wife, upon seeing me awake after 23:00, would say "Oh, are you gonna stay up?" or "Oh, you're staying up late!" or "Are you staying up late?"