First of: you should major in whatever you have the greater passion for. If you are passionate about aircraft or spacecraft, go for it. Things will fall into place. But if you're unsure, you might want to do mechanical engineering.
Mechanical engineering is the general engineering of dynamics, kinetics, fluids, etc while aerospace engineering is specialization of mechanical engineering. That means as a mechanical engineer you will learn fluid flows in pipes and ducts and in aerospace engineering you might focus on aerodynamics (fluid flows over wings). It is the same theory, but aerospace engineering applies it to aerospace specific problems.
Some schools actually have aerospace engineering as a focus in mechanical engineering rather than its own major.
Now, as for landing a job: usually the aerospace companies will accept either an aerospace engineer or a mechanical engineer (with a focus in the particular area of interest) for the same job. Engineering jobs are all competitive, no matter what field you go in. Making sure you will get a good job has less to do with your particular major and is more about working hard, doing internships or working in research labs, and applying your knowledge.
I work on inlet and jet engine design for highly integrated vehicles (that means an aircraft like the X43) so my colleagues are electrical/computer engineers, mechanical engineers, chemical engineers, materials engineers, aerospace engineers, physicists, mathematicians, computer scientists, chemists, and even some civil engineers. Electrical & computer engineers work on the control system of the vehicle, which is basically the brains behind the airplane. Mechanical and civil engineers work on various things, including the structural integrity of the airplane. Chemical & material engineers and chemists work on the fuel system and choosing the right materials for the airplane. Physicists work out the theory of the fluid flow which the mathematicians and computer scientists help develop into computer programs that help in the design of the vehicle. And aerospace engineers are kind of dispersed in all areas.
If I were you, I'd check out the school you are interested in and see both departments. Get a feel for which one you are more comfortable in, which ones have the more interesting research, and pick that one.
And of course, if you end up getting a master's degree, you will specialize even further.