Should I put an extra thick layer of Mastic in and around the joint and feather it out like you do with drywall joint compound, let it dry and try to sand it to smooth it out?
Or maybe just use drywall compound itself and Mastic over it once it's dried and
sanded?

If the drywall is 1/16" thicker then the durock just sand the drywall
down a bit with a palm sander to even out the transition. Apply some
mastic to the joint and lay some mesh tape over that to make a stronger
joint. Smooth a bit more mastic over the tape and let it dry.

Just decide before hand how far up you want the tile to go over the
existing drywall, a couple of inches maybe is enough. If you go to far
with the sander you will have to cover that up with paint or drywall
compound if it is bad.

This isn't nearly as bad as everyone is making it out to be.