Why do the front and rear brakes work independently?
I see several reasons:


Redundancy of an essential safety feature is good.  If there's a problem with one brake lever you still have the other brake. Being totally unable to stop could be disastrous.
It is a legal requirement in some places to have two independent brakes for a bicycle used on public roads.
Limited human hand strength. One hand can't pull the brakes as hard as two. If you need to stop really quickly this could make a difference.
Separate control of braking.  The rear brake is much less effective and prone to skidding. The front brake is more effective for quick stops, but locking up (skidding) the front wheel will lead to a crash. @Stephen Touset wrote up a detailed answer covering the control issues.
Allows signaling with either hand and stopping at the same time.  Sometimes you need to be slowing down to prepare for a stop or turn and signaling at the same time. If you can take one hand or the other (but not both) off the handlebars that can be useful. (I prefer the arm straight out in direction of turn signal instead of the left arm hooked up to signal right turn)
Simplicity of design.  A single lever controlling both brakes (via two cables?) would be tricky to adjust right, and if it wasn't adjusted exactly perfectly you'd probably end up braking with one wheel and the other never really happening. The longer cable (rear brake) would get more stretch (housing compression) just because it's longer, so you'd probably need to be fiddling with barrel adjusters on that cable quite frequently. One lever controlling both brakes just seems difficult to engineer right and likely to need much more frequent adjusting/maintenance.


Personally, I tend to use the rear brake for controlling my speed (slowing down just a little) and either the front brake or both brakes for stopping (especially a fast stop). An awareness at any given time of which tire is more likely to slip and using the other brake is good, though.

Worth noting: it's a specialty item, but you can get a brake lever that controls both brake levers at once.  This is mostly popular with bike polo players (who use one hand for a mallet) and people with physical conditions that limit their ability to use the brake with one hand. If you don't have any special reasons to need just one lever, you're probably best off with separate levers. http://www.paulcomp.com/duplexlever.html