Physically spoken, the processes are basically inverse of each other. XPS means X-rays in, electrons out while in EDS it's electrons in, X-rays out.

Both provide access to the stoichiometry by sensitivity factors, although both may require calibration.

EDS penetrates a bit deeper into the sample since the mean free path (or electron escape depth in XPS) is higher with the electrons' kinetic energies there.

Now, for two aspects you have certain advantages:

• chemical sensitivity/resolution: XPS has by far the higher resolution since peaks are narrower by a factor of ca. 100. Therefore XPS is suitable to analyze chemical shifts while EDS can't really help you with anything beyond stoichiometry.
• EDS is commonly installed in SEM or TEM devices and has the better spectromicroscopic resolution. This is essentially limited by whatever is the machine's spatial resolution at 10-15keV impact energy. For XPS, there is also an associated spectromicroscopic method called "SPEM" or "ESCA microscopy". In that case you use a Fresnel zone plate with which the smallest beam spot I've heard about is 80 nm, although I haven't done it in a while, so I'm not up to date on the state of the art.

It may be helpful to see these previous RG threads:

https://www.researchgate.net/post/what_is_the_difference_between_Xps_and_Edx_spectroscopy_techniques

https://www.researchgate.net/post/How-does-the-obtained-elemental-concentration-from-XPS-compares-with-the-EDX-results

https://www.researchgate.net/post/Why_the_At_of_element_in_SEM-EDS_is_different_from_XPS_analysis

XPS confirms the presence of the elements, their stoichiometry, and the atomic bonding between elements corresponds to different orbitals, while SEM confirms the morphology of the materials or the sample specimen.