A fulfilment service provides storage and distribution, larger ones will offer picking and packing as well. So if you are buying a large volume of products, but are perhaps not geared up to hold the stock or you are in another country and don't want to import, then this is where a fulfilment service is of value.
Of course the bottom line is they are dealing with your stock, so it's stock you have to purchase up front. A drop shipper requires no cash outlay for stock up front, at least that is in most cases. Some may charge membership fees or other type fees.
So with drop shipping you are advertising the product, then when you sell it you place the order with the drop shipper, and they ship it direct to your customer after you pay them.