At the university I attended, Purdue, there is an ME degree and AE degree, and while they do overlap, there are major differences. I didn't attend the main campus the first two years because I thought I could save some money that way, so my first two years were in ME (The local campus didn't have AE). However, when I transferred to the main campus I had to retake many of my sophomore year engineering classes. One specific example is in dynamics. In ME sophomore dynamics, we mainly concentrated on solving equations at a specific point in time. For Aero, we write the equations of motion for all time, then we just plug in for the time we want the results calculated.
Also, if you majored in AE, you have more AE oriented classes. I majored in aerodynamics and minored in propulsion. I remember having some ME's in my propulsion classes, but none in my aerodynamics courses. I also didn't have any ME's in my sophomore introduction to aircraft design course, or my senior aircraft design project course.
I would say if you plan to go into Aero, plan on getting an advanced degree. In fact, IIRC, if you wanted to get into the propulsion field, most of the students I knew weren't even planning to apply at the majors, P&W, G.E., and Allison at the time, if they didn't at least have a Masters degree.
For reference, I've spent almost all of my life doing ME work. So don't think because you go into Aero, you can't get a job as an ME either.
Mainly, I would say kind of what DSE said to you. Figure out if there is a specific area which you want to concentrate on. For instance, much in line with what DSE said, there was a lot of overlap at Purdue at the time between Propulsion and Thermodynamics. All of the thermo classes were taught by the ME department, but most of the propulsion classes were taught by the aero department. To a certain extent, I would say the Aero department really was just a more specialized offshoot of the ME department. While there was a lot of overlap in the basics, I don't think there was much in the way of orbital mechanics in the ME department, or as much fluid mechanics.
But think about what you want to do, or order them in number of most interested to least interested of what you would most like to do. Aircraft design/configuration layout? Performance analysis? Structures? Aerodynamics/Propulsion? System integration? Stability and control?
Also, the only place where my degree kind of held me back was when I worked for a couple of years in structural engineering. They prefer civil engineering degrees, because they are specialized for that field, but I did alright there and learned that there are a lot of architects who are really crappy at designing structures. But there were some who were actually very good as well, but they were rare.