The question was asked long time ago on this site, but not answered properly.
In classical thermodynamics, entropy is defined up to a constant. In statistical thermodynamics, there is no such freedom. What exactly is the reason for either point of view?
In classical thermodynamics, only entropy changes play a role. But why does this argument disappear in statistical mechanics?
Also several formulations of the third law allow for entropy to reach a constant value (non-zero) at lowest temperature.
The paper by Steane arxiv.org/abs/1510.02311 (he added it below) shows that the discussion is pointless, because absolute entropy values can be determined also in classical thermodynamics. He shows that classical entropy is NOT defined up to a constant. A great read.
The entropy implied here is observed entropy.