First off, who does it belong to anyway? Is it considered neither the property of the Federation, nor The People (Romulan, Klingon or other party) on the other side?
Is it in some way "joint property" of both parties?
Maybe the neutral zone is the modern equivalent of international waters, no one owns it, and while the Federation and the Empire can't enter it, third parties can employ it, work it, travel it, maybe colonize it, to their hearts content. Another possibility is that it's a "dead zone," no one can enter it, regardless of who you are, this would prevent the Federation/Empire from using mercenaries to conduct espionage and covert operations.
How big is it? Balance of Terror suggest that the RNZ really isn't too thick, a few light years. TNG didn't get specific, but it seemed to imply a greater thickness, something that took time to traverse. The Federation and Klingon Empire would seem to have a NZ only at certain points, in other places the Federation and Empire come together with a "normal" border. The NZ seen in the Kobayashi Maru test (Saavik's) looked more like a bubble, than a wall in space.
Does the existence of a NZ serve any useful purpose? Or is it just as good to have my territory come into direct contact with yours?