Bay is a redder color with dark mane and tail and legs.
Brown is - brown. Very little to no red in it.
In the USA, the terminology tends to be confused. There are a lot of brown horses that get called bay, if their legs or mane or tail are a little darker Technically, if it's a brown color, it's not a bay, regardless if the points are a little darker. Americans tend to call a lot of horses bay, that are actually brown.