Atomic mass
The atomic mass ("m") is the mass of an atom. Its unit is the unified atomic mass units ("abbr." u) where 1 unified atomic mass unit is defined as of the mass of a single carbon-12 atom, at rest. For atoms, the protons and neutrons of the nucleus account for nearly all of the total mass, and the atomic mass measured in u has nearly the same value as the mass number.
Atomic mass unit
The atomic mass unit (amu) without the "unified" prefix is technically an obsolete unit based on oxygen, which was replaced in 1961. However, many sources still use the term "amu" but now define it in the same way as u (i.e., based on carbon-12). In this sense, most uses of the terms "atomic mass units" and "amu", today, actually refer to unified atomic mass unit. For standardization, a specific atomic nucleus (carbon-12 vs. oxygen-16) had to be chosen because the average mass of a nucleon depends on the count of the nucleons in the atomic nucleus due to mass defect. This is also why the mass of a proton or neutron by itself is more than (and not equal to) 1 u.