I see people using these two words interchangeably. But I feel there is a huge difference between the both.
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".
There is indeed a fundamental difference between spaceflights and aeronautics. But historically, the space industry largely grew out of aviation, NACA became NASA, etc. There is a lot of shared knowledge and experience; many companies are active in both fields (Lockheed Martin, Boeing, etc.) An overarching term to cover both fields makes sense.
However, this is not a universal truth. In the USSR, for example, the postwar history went such that the space industry grew out of (and initially was managed by) the artillery. And although some aeronautics companies were later forcibly converted to space, there was no shared industry to speak of, and the term 'aerospace' was not in common use.