There is no difference. "Subway" and "Underground" are used to refer to the same thing in different places. Both refer to train systems. The train system in New York is referred to as "the subway". The train system in London is called either "the underground" or "the tube". The train system in Chicago is called "the el", short for "elevated", because many of its stops are above ground. In American English, we use "subway" as the generic term.