@albert I suppose in either case it's shorthand.
America is not a continent, there's "North America" and "South America," "The Americas" for both, I suppose...though that seems odd. The country isn't America but "The United States of America".
I suppose American could mean "Anyone from the two distinct but connected continents." but that doesn't seem terribly specific as a descriptor of someone.
It could also probably mean a descendent of Vespucci. :)