A big wing sump is just a normal sump with extra capacity. It is called what it is, because it has "wings" welded on to the bottom of it that carry the extra oil .

A dry sump is exactly what it sounds like, there is not oil actually retained in the sump, it is stored in a container located inside the car and is pumped around the engine from this. The benefits of this are a few bhp are gained from a reduction in drag, as the crank is not dragging through a large quantity of oil that it would in a "wet" sump.