A cooler might be a bit more casual...something you use around the barn at home or as a blanket liner. A cooler can also be one of those big square things, but not always. A dress sheet is never a big square thing.
But, I don't think there's really much of a distinction. Check out this description of the Rambo Newmarket Cooler from Beval's. The 2 terms are used to describe the same product.