Each region has a name for their transportation system, so it varies. For New York, I have mostly seen "subway", for DC it's "metro", for Boston and Pittsburgh it's "the T", for Chicago it's "the L".
In terms of strict dictionary definitions, both terms imply underground trains. Subway is a generic term, but metro is more localized to certain cities.
From M-W:
subway: a system of underground trains in a city
metro: an underground railway system in some cities
