Both are numerically same! Yeah, You have read it correct. ;-)
But yes there are two differences you should be aware of.
Difference 1. The Units
We generally use amu ( atomic mass unit) or u (unified mass) while talking about mass of atoms, molecules and ions.
On the other hand, We generally use gram (g) per mole when we talk about molar mass.
NOTE : 1 amu=1u = 1.67 * 10^24 g
Difference 2. The Meaning
Mass of an atom is called Atomic mass. And this is the mass of 1 atom of that element (like Hydrogen).
For example the mass of 1 atom of H is 1 amu.
Similarly, Molecular mass is the mass of one molecule of that kind.
For example Molecular mass of Ca(OH)2 is 74 amu.
Molar mass is the mass of 1 mole of molecules of that element.
For example, Molar mass of hydrogen is 1 g, This means that 1 mole of hydrogen atoms will have mass around 1 g.
Note: In many formulae,, you may find the following terms interchanged:Atomic mass , Molecular mass, Formula mass and Molar mass.For quick response, just remember that molar mass is atomic mass or molecular mass in grams.