Which is a better unit for expressing molar mass, amu or grams/mole?

While the numeric values are the same the units convey different information.
The term amu refers to the unit which is the value of a proton and the value of a neutron in regards to a mass comparison to the standard that carbon – 12 has 6 protons and 6 neutrons.
When the designation for the the mass is ##(grams)/(mol)## the unit refers to the fact that in 1 mole of a substance there is a specific gram amount.
Let us look at an example of Carbon Dioxide – ##CO_2##
The gram mass of of carbon dioxide ##CO_2## can be determined by combining the mass of one carbon atom and 2 oxygen atoms.
C = 1 x 12.01 = 12.01
O = 2 x 15.99 = 31.98
12.01 + 31.98 = 43.99 grams
This value can be designated as ##(grams)/(mol)## meaning that in one mole (##(6.02 x 10^23)## molecules) there would be 43.99 grams of ##CO_2##.
However, if the unit is designated as amu the information informs us that there are 6 protons + 6 neutrons = 12 ma in carbon and 2(8 protons + 8 neutrons) = 32 amu in oxygen for a total of
44 amu in a ##CO_2##.