Molar mass is the mass of a mole of a substance, whether it is an element or a compound. The mole is a unit in the International System (mol).  It  represents the number of atoms or molecules.  This number, known as Avogadro’s  Number, is enormous:   N_A =  6.022×10²³. It is perfectly adapted to chemical calculations.  It is clearly easier to  manipulate .5 moles of atoms than 300 thousand billions of billions of those atoms, even if the these two numbers represent the same thing.

• Atomic molecular mass is the mass of a mole of atoms.
• Molecular molar mass is the mass of a mole of molecules.