Avogadro's number is an experimentally measured quantity. Although we are confident that we know its value quite well, some future experiment may cause us to make a small revision in the number. By necessity, the mass of 1 m in grams will change accordingly. This is not worrisome, because neither number is crucial to the utility of the mole. The atomic weight plays a dual role in chemistry.
Many chemists prefer to use the term molar mass for the mass of a mole of substance.
In this course, we will use the phrase Formula Weight for both situations. The meaning will be evident from context. This is true because 1 mole of substance contains N A atoms, molecules, or formula units of substance. 5.6 g of iron is not much iron. However, even this small amount contains a huge number of iron atoms.