A lot of people use these three terms interchangeably. Restaurants, bistros and brasseries to many mean only slight differences and astonishingly, many think that the difference lies in the size of the establishment. Here’s a little backstory (much of which is taught in schools) to go with the classifications and hopefully this can clear up the confusion.
To start with Restaurants are always formal whereas Bistros tend to be more casual with less bells and whistles. Bistros do tend to be smaller with a small kitchen and staff and a simple menu, though it would be wrong to classify them as such. Some bistros with passionate owners tend to serve impressive menu items but then again it all boils down to the owners and each individual establishment.
In fact, if you remember the Bistronomique movement and the neo-bistro movement, then you know that the concept revolves around cooking exquisite delicious menus in the simplest of environments. Basically, bistros are a place where food is the sole priority, not the environment, sitting arrangement, dress code and other things.
As for Brasseries, they were typically a place where beer was served. The dishes served in such establishments tend to be of the local variation with nothing concrete. Brasseries are wide, large and tend to operate all day long. Nowadays though brasseries tend to concentrate more on wine.
On the other hand, a restaurant is more generic. Naturally, it also happens to be the easiest word to understand. Unlike a brasserie or bistro where the impetus is never on the food, in a restaurant it truly is the only thing that matters. You can have a splendid surrounding, amazing beverages, serve the best wine but if your food menu is subpar then you are not a good restaurant. Although, presentation and environment does count to a certain extent in ratings for a restaurant.