This is really a question about verb tense, and I'm surprised no one has really given an answer that clarifies the difference between the two in general. English has present tense (sleep) but also a "present imperfect" tense (sleeping). They are very similar, but present imperfect is for actions that are continuous, or unfinished.
It's probably more common than present tense, honestly, especially when talking about things people do, because they can be changed. If I'm talking whether the plumbing in my house works, I might say "the water runs" because at the current moment, it can, and if I turn it on, it does. But if I'm talking about accidentally leaving the faucet on, I'm more likely to say "the water is running" because it is a present state that is likely to change at some point but has not yet stopped.
In the case of "you can sleep while I (drive / am driving)", as others have pointed out, the meaning doesn't really change. I do believe there is a very subtle difference though, which is whose action the tense puts the focus on. "While I am driving" is a continuous, unfinished activity, whereas "while I drive" is more of a discrete single-event action. So, to me, "you can sleep while I drive" puts the focus on the sleeping person, because to them, the driving happens while they're asleep with no observable time elapsing during the drive; but "you can sleep while I am driving" puts the focus on the driving person, because to them, they have to make a continuous effort that won't be finished for a while.