Great question! In general, astrophysicists would work more on the questions like "how does the universe work?" and "how did we get here?" and "are we alone out here?". They are scientists who look at data gathered by satellites, mathematical models, and work in observatories. They may veer into more applied areas, like creating sensors that will help measure a parameter, or they may stay in very theoretical realms, looking at models.
Aerospace engineers are usually much more on the applied side of how to actually get that satellite into space. They work to make bigger, more efficient rockets or smaller, tailored rockets. An aerospace engineer might be responsible for coming up with the trajectory (or the path) the satellite will take and make sure there is enough fuel on board to carry out the mission. They help design and build thermal systems and materials that can handle the extreme heat and cold. Basically, anything relating to the design, build, launch, and operation of a satellite would be mostly aerospace engineers.
I also feel I should point out that aerospace engineers can also work on things not related to space. We work on aircrafts, jet engines, helicopters, submarines, atmospheric research... we do a broad range of things that are not even related to space; something to keep in mind.
Aerospace engineering is usually its own degree; some schools combine aerospace and mechanical engineering in one department or offer a mechanical engineering degree with an aerospace focus. Astrophysics might be its own degree or be a area of specialization within physics departments, but it will depend on the school. Most big state schools that offer aerospace engineering would also offer a degree in astrophysics and/or physics. Smaller, liberal arts schools will often offer a physics degree but might not have an aerospace engineering degree. So it really is up to where you want to go.
The University of Colorado in Boulder offers both an astrophysics and an aerospace engineering degree, and both are highly ranked. The University of Maryland, College Park also offers both degrees and I can attest their aerospace engineering program is great (my alma mater). MIT, Cal Tech, Stanford etc all have great programs for both. I would recommend a larger school so you have the option of switching majors if you so choose.
Best of luck to you!