I haven't tried the Hyatt place yet, but comparing courtyard and HGI, I find ghi's usually newer and rooms are nicer. Internet usually is more reliable too rather than the Stsn thing at Marriott's.
Also, I like the food (breakfast) better at HGI, esp. The free breakfast for gold and diamond - works great when traveling for personal reasons without a spending account.
I also find Marriott's site more convoluted (could be just me).
Not sure if I noticed any staff difference.

I don't I paid that much attention to that side of things. Never had issues as a hh hold or mr silver.
Anything I missed?