To use a computer analogy, sleeping and particularily REM is your brain's way of running a defrag program. All the information your mind has collected or created is organized.
Also, sleep is your brain's time to balance out it's serotonin levels and replenish its serotonin. Serotonin is very important for mood regulation, several forms of cognition and memory, and digestion.

So essentially, without sleep you'd go nucking futs.