Aeronautical engineering is, in short, the study of aircrafts. The study of airfoils (wings) and propulsion (which includes engines) are two main components that make up the aeronautical field. Airfoils help the plane counteract the force of gravity. They can also be used to help the aircraft change directions. Without propulsion the aircraft would have a significantly smaller amount of air time because there would be no force to keep the aircraft afloat. Because the basic dynamics of aeronautical engineering are putting an object into flight and keeping it there, there is the constant material issue of strength verses weight. The object must be made of strong enough materials to be able to withstand the forces pushing against it, but it must also be light enough to have the ability to fly through the air.
Astronautical engineering is the study of flight and navigation outside of the Earth's atmosphere. One of the biggest concerns in the astronautical field is the maximizing of payload, the carrying capacity of the spacecraft, while keeping the mass and weight of the spacecraft as small as possible. Astronautical engineers must find ways for the spacecraft to withstand extreme temperatures (and extreme temperature changes), high-grade vacuums, and magnetic belts. They must also be conscious of the changes space will have on elements of the spacecraft. For instance, fluids have to be confined and extruded, liquids my evaporate or freeze, and heat must come from radiation.
Both aeronautical and astronautical engineering fall under the larger category of aerospace engineering. Both branches are concerned with the flight and navigation of an object and both require intelligent, creative people to pave the way to a new tomorrow.