Atomic mass
The atomic mass or relative isotopic mass refers to the mass of a single particle, and therefore is tied to a certain specific isotope of an element. The dimensionless standard atomic weight instead refers to the average (mathematical mean) of atomic mass values of a typical naturally-occurring mixture of isotopes for a sample of an element. Atomic mass values are thus commonly reported to many more significant figures than atomic weights. Standard atomic weight is related to atomic mass by the abundance ranking of isotopes for each element. It is usually "about" the same value as the atomic mass of the most abundant isotope, other than what looks like (but is not actually) a rounding difference.
Atomic mass constant
In practice, the atomic mass constant is determined as the ratio of the electron rest mass "m" to the electron relative atomic mass "A"(e) (that is, the mass of the electron on a scale where C = 12). The relative atomic mass of the electron can be measured in cyclotron experiments, while the rest mass of the electron can be derived from other physical constants.
The current uncertainty in the value of the atomic mass constant – one part in 20 million – is almost entirely due to the uncertainty in the value of the Planck constant.