It would perhaps be more accurate to use the terms formula mass and molecular mass rather than formula weight and molecular weight. This because in physical terms, weight is a force while mass is a measure of the amount of substance in something.
Both formula mass and molecular mass refer to the mass of a compound relative to 1/12 of the mass of a Carbon-12 atom. However, molecular mass is specific to molecules – that is, only for a minimum of 2 atoms held together by covalent bonds.
As an example, you can say that the molecular mass of water is 18. You can also say that the formula mass of water is 18. You can say that the formula mass of common table salt, NaCl, is 58.5, but it would be inaccurate to say that the molecular mass of NaCl is 58.5, since NaCl is not a molecule.
The difference is not in numerical value but merely terminology.