What exactly is the significance of molar mass in chemistry? Why not just mass?
molar mass of a substance is the mass of one mole of a substance
This is not quite correct. If you said the speed of a car is the distance it travels in one hour, cars would go at speeds of 120 miles (race car) or 30 miles (in town). However, speed is defined as distance traveled per time. As a result, speed has the dimension of distance divided by time, and a typical unit (at least where I live) is miles per hour.
Molar mass is defined as the mass per amount of substance. The dimensions are mass divided by amount of substance, and typical units are g/mol.
Why does molar mass have the same value as molecular mass?
If the molar mass is given in g/mol and the molecular mass is given in u (unified atomic mass unit) or Dalton, the number is the same. The reason is that both the mole (at least until May 2019) and the Dalton are based on the mass of the carbon-12 isotope. A mole of a substance is defined as the amount that has the same number of particles as present in 12 grams of carbon-12. The Dalton is define as 1/12th of the mass of a carbon-12 atom.
From these definitions, you will find that the molar mass of the carbon-12 isotope is 12 g/mol, exactly, and the molecular mass of carbon-12 is 12 Dalton, exactly.
What will change in May 2019?
The mole will be defined as "One mole contains exactly 6.02214076×10^23 elementary entities." The molar mass of carbon-12 will be an experimentally determined quantity.