There is a whole evolutionary "theory" about it by the holder of the current sleepiness record (although like Ed noted, the guy seemed to be going to sleep most of the time of the experiment).
Without buying too much into this theory, my modest experimentation totally confirms two things:
1. I do have bursts of creative thought as soon as I deprive myself of a little bit of sleep (and I do enjoy those a lot!, can't deny healthy natural highs like this ;)
2. The full potential of the brain can be mostly accomplished by suppressing processes instead of creating new ones. This is a confirmation of Marvin Minsky's theory of the mind which I deeply admire. More on this to come on future post as I advance on his latest book about emotions.
General consensus on what the two brains are about:
Looks at parts
Looks at wholes
The parts vs. wholes thing reminds me of a story my friend Skooby told me about a book he read describing an experiment that made evident how different occidental and oriental perceptions on a same experience are. Maybe the explanation on this could be that orientals are more right brained and occidentals are more left brained ...
Update: Also it's said that music connects both hemispheres ... I like to believe this is true.