There is a lot more to think about on top of what I posted, concerning the strange nature of sleep itself.
The latest theory is that sleep is an "emergent property" of the brain, meaning it's a result of many different sort of unrelated events, like different subsystems of the brain all shutting down at the same time. One consequence of this is that even if you don't give them a chance to go down all at once with regular sleep, some of them will shut down and go to sleep independently, while you're still basically conscious. You've probably experienced some form of this if you were ever woken up from a deep sleep and you couldn't talk right away or you were confused and felt all weird.
Anyway, that sort of thing is responsible for reducing your performance the next day, parts of your brain will literally be asleep.
Then your brain goes into compensation mode. It compensates by increasing your dopamine levels, which makes you feel better than you should, and helps make you a little more alert.
I've found that to be a really noticeable effect, almost a high. Last summer when my girlfriend practically lived with me, she didn't get off work till 10pm and I woke up to go to work at 6am. As a result I didn't sleep much at all, and I'd feel really goofy and happy during the day. Other people perceived me differently though: for example, when my boss and I went out for coffee one day I said something, I don't remember what, to which he paused and thought for a while, and then broke down laughing. "An iz... you say the STUPIDEST shit sometimes man..." and he literally leaned against a building lauging his ass off while I was just like "what..?"