"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.