Having spent an embarassing amount of my life in American big box retail I am about to get really, really nerdy about this and yet also not really define anything either.
Grocery store and supermarket are mostly interchangeable terms. It's one of those things like Coke vs. soda vs. pop, the preferred term depends on where you are. Whichever word you use, people will generally not be confused. Again this is from U.S. perspective.
However, it's also one of those situations where every square is a rectangle but not every rectangle is a square. There are "grocery stores" which are not supermarkets. They tend to be smaller stores, are more likely to be independent stores or part of a small regional chain (not part of a large chain like a supermarket typically is), and often, but not always, specialize in a particular kind of food or cuisine and often sell products you can't typically get in a regular store. Two examples:
- Near where I live, there's an organic grocery store that specializes in locally produced foods and organic convenience foods. But this store is very small, they only sell a few products and you couldn't go there and buy all your food for a week unless your eating habits are very peculiar.
- Where my extended family lives, in a small rural town, there's only one store that sells food, and it's relatively small, and it sells very basic staple foods, and no one calls it a supermarket. But you can buy a week's worth of food there easily enough.
Wal-Mart, at least when I worked there, called itself a discount retailer and saw groceries as one of several "lines", each composed of different departments. The store I worked at considered itself a Wal-Mart store, but it considered the grocery departments (freezer, bakery, meat, produce etc.) together to be the store's "supermarket". So per Wal-Mart, the store it was not a supermarket but it had a supermarket.
I would think most people would call Wal-Mart a discount retailer, a general retailer, or a discount department store, but some people would call it a supermarket and none of those people are wrong honestly.