Comparing these two wars, however, may be comparing apples to oranges. I do think it gives an interesting perspective though.

Both are used to refer to the same thing. Basically, anyone who fought for the United States during the war.

The closest thing you'll find in terms of a difference is "volunteers" and "regular Army". As time went on though, the regular army played a smaller and smaller role. I read somewhere that during the Mexican-American war, volunteers made up roughly 66% of US forces. During the civil war, it was 97%.