Sometimes I stay awake so long I forget I'm tired, yeah. Especially the case when I suddenly have to focus on a task that's not a lazy one, such as say, cooking.
You generally don't gain anything when you sleep in terms of energy, though. Literally nothing in your body changes, it's all your brain. Sure there are theories your body does all sorts of repairwork inside, such as reparing stuff the bacteria might eat away that you can't feel, or just fixing and kind of massaging muscles by flexing them in the night and so on, and also unwinding your skeleton and stuff but..it's all up in the air really.
My theory is sleep simply lets your mind defragment what you've learned, seen and done today. It deletes 80% of it and keeps partial parts of all of it, making up about 20% of what you saw. Which then degrades over time anyway. Anyway I guess what I'm saying is the sudden rush of freshness is your mind having been forced to do some background recalibrating since you didn't go sleep, and then suddenly finished sorting a part of your mind that was working with all the stuff you've done recently. Once the cache was cleared, so to speak, you suddenly feel fresher - if only for a while, until you see more stuff and the burden raises up again, and makes your mind tell you you need to sleep so it can work in **** peace without you adding more to the pile.
Saw an interesting advert once to do with cameras, a guy was walking through a forest and everytime he blinked, an image snapped out of his head (I think) and appeared on the ground at exactly what he was looking at. Perhaps the mind works in a similar way, taking mini pictures constantly, which then need to be deleted or sorted while you sleep.
Did I mention I'm crazy? Blame that if I make no sense.