Hey.

Aerospace (aero-space) is a superset of aeronautics (atmospheric flight) and astronautics (space flight). An aeronautical engineer is an aerospace engineer, but not necessarily vice versa.
Aeronautical engineering purely deals with the study of objects that stays within the earth's atmosphere such as airplanes and helicopters, and its property. aircraft designing is the major area covered under aeronautical engineering . Universities providing aeronautical engineering degree make students to learn

Aero Engineering Thermodynamics

Heat Transfer

Aircraft Structures

Flight Mechanics

Aircraft Stability and Control

Aerodynamics

Aerospace engineering deals with study of aircraft that operate outside of the earth's atmosphere such as rockets, satellites, and other and spacecraft.Universities providing aerospace engineering degree make students to learn

Spacecraft Design

Introduction to Astronautics

Fundamentals of Jet Propulsion

Spacecraft Dynamics

Orbital Mechanics

Rocket Science

Many foreign universities Provide aerospace engineering course , which is a combination of aeronautics and astronautics .
Both Aerospace and Aeronautical Engineering are good branches. You should choose the branch depending on the career path you foresee for yourself. If you want to work in the aircraft building industry then aerospace engineering is the right course for you. However, if you look forward to work in the space industry, then aeronautical engineering is the right branch for you.

You can also choose aeronautical specialisation after pursuing B.Tech in Aerospace or Mechanical Engineering. Make sure that you take admission in good colleges while choosing these streams of engineering in order to ensure jobs.

Hope this helps. All the best.