Materials. Closed cell foam is any foam, of any material, that has closed cells. That is, it's got little bubbles of gas, usually plain air, in it, that are distinct from each other. It's water proof, doesn't absorb materials, etc. Open cell foams have bubbles that are largely connected to each other. It'll absorb whatever it's in contact with. A sponge is an open celled foam, typically of cellulose. So are the foams in some sorts of self-inflating mats.
The difference between neoprene and 'ordinary' closed cell foam is one of material. Closed cells foams are normally, in our context, polyethylene, some times polyurethane. Different materials have different properties.