Aeronautical Engineering

Aeronautical Engineering can be said to be a part/section of Aerospace Engineering. Aerospace Engineering is all about manufacturing, planning, maintaining and designing Aircraft and Spacecraft. Aerospace Engineering can be said to be made of two sections- Aeronautical and Astronautical Engineering.
During the first year, common engineering subjects are taught. Some important first-year subjects are:
Engineering Mathematics (1 and 2)
Elements of Electrical Engineering
Elements of Civil Engineering
Engineering Graphics
Engineering Physics
Communication Skills
Computer Programming Basics
Elements of Mechanical Engineering
Mechanics of Solids
Environmental Science
*The first year consists of semesters 1 and 2. From 3rd semester onwards, subjects focusing on the field of Aeronautical Engineering will be introduced. Some such subjects present in the 3rd semester are-
-Engineering Mathematics (3)
-Fluid Mechanics
-Introduction to Thermodynamics
-Basics of Aeronautics
-Strength of Materials
-Machine Elements
*Now, let us check out some important subjects that one will come across in the 4th semester.
-Engineering Mathematics (4)
-Aircraft Structures (1)
-Aerodynamics (1)
-Propulsion (1)
*Important 5th-semester subjects-
-Aerodynamics (2)
-Propulsion (2)
-Flight Mechanics
-Aircraft Structures (2)
*Important subject in 6th Semester
-Theory of Vibration and Noise Control
-Avionics
-Introduction to Control Theory
*Important 7th-semester subjects-
-Aircraft Design
-Theory of Heat Transfer
-Computational Fluid Dynamics
-Aircraft Systems and Instruments
*Important 8th-semester subjects
-Space Dynamics
-Rocket and Missile Design
-Aircraft Maintenance
-Control and Navigation
These were some important subjects that one will come across, while pursuing BE/B Tech Aeronautical Engineering program. Please note that in case of some subjects, along with the theory part, students will also have to go through lab/practical sessions also.

Aeronautical Engineering deals with the designing of new aircrafts that are limited to the earth's atmosphere and deals with the concepts, components and forces that affects the aircraft.
For example- aerodynamics, material Science and propulsion.
Aircraft maintenance engineering deals with the repair of these aircrafts. This can be part specific.
So in comparison, Aircraft maintenance engineering deals with repairing existing aircrafts whereas aeronautical engineering deals with creating new and improved aircraft.
This is pretty base answer to your question, but is a good start. Anybody else can feel free to expand knowledge on this.
Have a wonderful day ahead.
Thank you.