Firstly, aerospace is a vast domain. It is subdivided into aeronautics (aeronautical engineering deals with this) and astronautics.

Now, coming to your question, aeronautics and astronautics have common fundamental concepts which includes propulsion, structures, aerodynamics and other subjects. These make sure that the fundamental concept remain same for Aerospace and Aeronautical engineering.

Aeronautics is all about dealing with aircrafts which are operational within the atmosphere like aeroplanes, helicopters etc.

Astronautics is all about dealing with aircrafts which are operational beyond the atmosphere (also called spacecraft) like space shuttles etc.

So, the verdict.

Aerospace engineering deals with both aeronautics and astronautics.

Aeronautical engineering deals with only aeronautics.

Now you make a choice.

Hope you find it helpful

let me know in the comment box if you are having any further doubts

thanks

Good Luck!