The terms are often used interchangeably and arguably context is the only functional differentiation in my experience, but I think the periodic table displays what should be called the atomic weight (the weighted average of the atomic mass of all natural isotopes of that element) and atomic mass is specifically the mass of a single atom.
For example, carbon has an atomic weight of 12.0107 (from memory, might be off a bit) and a carbon-6 atom has an atomic mass of 12.
Source; Masters degree in Chemistry, albeit over a decade ago now
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