Atomic mass
The term "atomic weight" is being phased out slowly and being replaced by "relative atomic mass", in most current usage. This shift in nomenclature reaches back to the 1960s and has been the source of much debate in the scientific community, which was triggered by the adoption of the unified atomic mass unit and the realization that weight was in some ways an inappropriate term. The argument for keeping the term "atomic weight" was primarily that it was a well understood term to those in the field, that the term "atomic mass" was already in use (as it is currently defined) and that the term "relative atomic mass" might be easily confused with "relative isotopic mass" (the mass of a single atom of a given nuclide, expressed dimensionlessly relative to 1/12 of the mass of carbon-12; see section above).
Molar mass
The precision to which a molar mass is known depends on the precision of the atomic masses from which it was calculated. Most atomic masses are known to a precision of at least one part in ten-thousand, often much better (the atomic mass of lithium is a notable, and serious, exception). This is adequate for almost all normal uses in chemistry: it is more precise than most chemical analyses, and exceeds the purity of most laboratory reagents.