In the US, we are loose about the term "city." The "city" I live in has a population of I 76,000. The neighboring town, Fort Collins, has a population of 164,000. But in 1990, my town, Loveland, only had 40,000 residents and I still would have called it a city.
The term "big city" is more specific. That's a major city, probably with skyscrapers. Population is probably close to a million or more.
The term "metropolis" is even more specific. That's a really big city, one of the top 20 in the US, to give a rough sense. Chicago, New York, Los Angeles are metropolises. Denver? It's a big city, but I don't know about metropolis.
The term "big city" is more specific. That's a major city, probably with skyscrapers. Population is probably close to a million or more.
The term "metropolis" is even more specific. That's a really big city, one of the top 20 in the US, to give a rough sense. Chicago, New York, Los Angeles are metropolises. Denver? It's a big city, but I don't know about metropolis.