Because of their noisy engines the AWACS aircraft are not allowed to fly in America anymore, so why can they still do so here?

This idea is wrong. Neither in America nor in Europe is there a “flight ban” applying to AWACS aircraft. Military airplanes are not subject to the same standards as commercial aircraft, just as the standards for trucks using the roads are different from those for passenger cars. NATO AWACS aircraft can be operated anywhere in the world - they regularly make use of civil airports such as Munich and Berlin, and every year they take part in exercises in the U.S.A. and Canada.