What is the real difference between aerospace and aeronautical?
Aeronautics is a subset of aerospace.


  Aerospace is the human effort in science, engineering, and business to fly in the atmosphere of Earth (aeronautics) and surrounding space (astronautics). 
  source: https://en.wikipedia.org/wiki/Aerospace


They are, somewhat incorrectly, used interchangeably - to colloquially refer to the broad subject of "flight".