In the United States, the word "subway" always means an underground train. We would never say "underground train" but we would understand it. Often we use the specific name of the local system; in Washington, D.C. we would more likely say "the Metro" instead of "the subway," and is Boston we would say "the T" instead of "the subway." (I believe that in London they talk about "the tube" or "the underground" and a "subway" is pedestrian tunnel). To confuse things completely, there is an historical term, the "Underground Railroad." It was not a railroad and was not underground; it was a secret system of routes and safe houses by which African-American slaves escaped from the slave states into free states and Canada.

They mean the same thing. I hear subway in the United States, particularly New York City. I hear underground and also the tube in the United Kingdom.

They both have the same meaning, however, "underground railway/train" is commonly used in the UK while "subway" is used in other vicinities such as the United States, etc.