You mention that if you reinstall the old outlet the both the dishwasher and other outlet work fine including the outlet that you are replacing with a GFCI.
If you've hooked up the dishwasher and GFCI and it works fine AND removed the dishwasher and hooked up the outlet and GFCI and the outlet works but, the GFCI works then it must be the oulet.
However, why would this oulet work with the old non-GFCI and not work with the new GFCI?
I say hook your dishwasher and outlet to the LOAD side of the GFCI and the check for power to the oulet that is not working.
Pull that oultet out of the box and put a meter on the wires going into the outlet. It might be that you have power to the outlet but, not coming out of the outlet in which case you have a bad outlet. Why this oulet will work with the non-GFCI outlet is beyond me but, at this point you have yet to pull the outlet in question.
Electricity is highly predictable so I'd say you have something weird going on here that we are unable to determine without being there.