To a certain extent it is a matter of personal preference. However, an advantage of using a pizza stone, especially one made from a refractory material, is that the stone will absorb moisture from the dough and produce a crispier crust. You will also usually get a good oven spring as the dough hits a stone with its high heat capacity (which is much higher than the surrounding oven air). This makes a stone especially good for a NY or Neapolitan style crust; however, for best results the dough should not have a lot of sugar in it because the sugar will caramelize too quickly and can cause the bottom crust to brown up too quickly and excessively, often before the top of the pizza is finished baking. From my experience, this makes a screen a better choice for a high-sugar dough, such as many American style doughs. A screen is also a good choice for me here in Texas where I don't want to heat up my stone for an hour, especially in the dog days of summer. To get up to the desired ambient oven temperature, say, 500 degrees F, it only takes about 10-15 minutes. Another benefit to a screen, of course, is that you dispense with the need for a pizza peel and navigating a pizza onto a stone.
I often use a combination of both a screen and a stone, especially when I want to make a pizza that is larger than my stone. I bake the pizza on the screen until the top crust starts to turn brown and the cheeses start melting, and then transfer the pizza onto the stone (preheated) for final bottom crust browning and crisping. That works real well.