At this point there are two answers: one says that there is no difference, except for the units; the molecular weight (MW) is the mass of one molecule, while a molar mass is the mass of a mole (6.02 x 10 ^ 23) of molecules.
While in certain contexts both statements can be shown to be true, I'm not sure these answers really provide you with the insight you're looking for.
So, we'll look at your question from a different perspective. Take a real example, something simple like water. Water has a nominal MW of 18 (so we'll use nominal precision for simplicity, instead of the 4-places often used for these types of calculations).
The MW of water (formula = H2O) is the weight of one atom of oxygen, which = 16 amu [8-neutrons at 1 amu each plus 8 protons at 1 amu each = 16 amu), plus two atoms of hydrogen, at 1 proton (1 amu) each. Normal everyday hydrogen has no neutrons. So for H2O, we have a total molecular weight of 18 amu.
From the CRC reference book we find that one amu weighs 1.66 x 10 ^--24 grams. Multiplying the two and in keeping with the two units of precision we're working with, one molecule of water has a mass of 29.8 x 10^-24g, or [3.0 x 10 ^ -23 g] per molecule of water.
A Mole is simply Avagadro's number (6.02 x 10 ^ 23) of anything... protons, baseballs, whatever. The term Molar Mass in chemistry refers to the mass of a mole of molecules. So in this case a molar mass of water molecules is Avagadro's number of them, the mass therefore being [6.02 x 10 ^ 23] x 3.0 x 10 ^ -23 g/ molecule] = 18.0 g
Summarizing:
Molecular Weight is the weight given in amu of an atom or molecule. For H2O, the MW is 18 amu or 3.0 x 10 ^ -23 g.
Molar Weight is the weight, usually in grams of 6.02 x 10 ^ 23 measurements which happens to be equal to the MW of the molecule (or atomic wt. of the atom) , and for water is 18.0 g.
So, while MW and molar weight are related, their absolute values are magnitudes apart.
