Genghis,
I only half answered your question.
I would say the difference is mostly historical.
In the days when it was still called aeronautical engineering, the curriculum didn't really have anything concerning space flight (such as orbital mechanics, launcher structures, guidance ...). If you wanted to specialize in that field, it would be post-grad, even if you could fall back on subjects like structures, aerodynamics and stability and control from your earlier courses.
Once space-flight-related subjects became part of the regular curriculum, it became more logical to call it aerospace engineering (and also it sounded more "advanced", of course ....).
CJ