Metropolitan is usually the city and it's suburbs.
So it's like Los Angeles population is a certain amount but if you add in its suburb which are also there own cities (Hollywood, Beverly Hills, Pasadena etc..) than that is the Metropolitan population
City population is the amount of people living in the actual city (for example, NYC has a population of 8.1 million).
Metro population is the amount of people living in the actual city PLUS the immediate surrounding cities/towns that comprise the METROpolitan population.
New Delhi is a part of Delhi and New York City is a part of New York so the metro counts include people who don't actually live in the cities, but in surrounding areas (most likely other parts of the state or close to it) that are within traveling distance and are likely to travel into the city for work and whatnot.