I understand that Auger electrons are emitted when an electron falls from a higher energy level to a open inner shell and EDX uses electrons that fall from outer shells when an inner shell is kicked off. I also have read that AED has a higher spatial resolution but I don't understand WHY this spatial resolution is higher. What is different about the Auger electrons that increases resolution?
I think this is a good question. The problem is more fundamental. In both cases, you would be using electron beams on the sample.
In EDX, you are shining a beam of electrons and detecting the X-rays. Here comes the main problem. It is hard to focus the X-rays. Mea culpa here. As Jon commented it is the depth of penetration of the electron beam and the corresponding X-ray emission that main affects spatial resolution. This picture roughly tells the story:
Auger electron electron spectroscopy, you are focusing a beam of electrons and you detect electrons. It is relatively easy to focus electrons using "lenses". Hence you can point point the exact origin where the electrons are being ejected.
Also, think about the depth profiling. EDX is not a surface technique unlike Auger. I used to have this misconception until I attended a workshop on surface analysis. The electrons can easily penetrate the sample and similarly originate from the some depth as well.