Not the same parts are active, though.
Sleep gives the body a chance to repair itself and for the immune system to fight infection.
http://www.meaningofdreams.org/dream...m#whydowesleep
Dreaming is also essential, and it wouldn't be safe for us to walk around while we were dreaming; we need to be immobilized.
I find it interesting that predators sleep much more than herbivores - think of how much time a cat spends sleeping.
Of course, this is partly because herbivores need to stay awake and alert as much as possible, to avoid becoming prey, but I also think that hunting for food requires a higher level of mental ability, and sufficient sleep is needed to bring it up to the necessary level.