I'm not sure what a black box is, but I know we have a procenium. To try to explain the difference is hard just because its a very visual thing. Our procenium is basically a wall, but the edge of a wall. You have your main theatre space that has all the battons over it, and then a section in the front closer to the audience that just has the theatre's ceiling above it. Normally there is a large arching wall or something that seperates these two spaces. That wall or the inside edge of the wall is the procenium as I have always known it. Thats kinda the best I can say, hopefully someone will come in and post a picture.
It's less about the benefits and more about the different purposes. As mentioned earlier, a black box is a much more intimate space. It's more suited to small plays, drama, etc. Bad college theatre, things like that. Typically in a small room like "guest" said. There is typically a very low lighting grid that covers the entire room to allow for lots of experimental possibilities.
A procenium is more like a hole in the wall. There is usually not much stage that protrudes beyond the arch, and there is (hopefully) a good deal of wing space on either side, and fly space above.
Guest also mentioned a thruststage, which is a bit like a procenium, but usually has a good portion of the stage jutting out and surrounded by audience on three sides.
That makes sense. I have only worked with "procenium" type stages. We have built a thrust on one of them, and on another i consider it a "black box" based on your description. this helps make things much more clearer