A metropolitan area is a large area of developed land, which surrounds a city, and is often made up of many buildings. It can include smaller towns and villages which are now connected to the larger city as it has grown. The biggest such metropolitan area is that of Tokyo. 8.5 million people live in Tokyo itself, 36.5 million live in the metropolitan area.
Looking from the air down to a populated area, there will often appear one big "blob" of buildings and lights. The entire built-up area is the metropolitan area. The city is often just the part near the center of the "blob". However, it may not be in the center. Sometimes there is even more than one city inside the "blob".
So to put it simply, looking from the air again, the "blob" is the metropolitan area. The city is a smaller section of the "blob".