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The only difference is the name of the unit; the numerical values are the same. However, the value in amu is the mass of a single molecule, while the molecular weight in grams is the mass of Avogadro's number of molecules.

The mass of Avogadro's number of molecules is called a Gram-mole. The molecular weight of , say, Hydrogen H2 is 2 amu. A gram mole of this would be 2 grams. The molecular weight in grams is the weight of this molecule in real grams and is an extremely small value.

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