Consider how entropy is mathematically defined in classical thermodynamics:
$$\delta Q=TdS \;,$$
in which $\delta Q$ stands for an "element" of heat, transferred from a classical source to a classical destination (which are classical, continuous substances), $T$ for the temperature of both the classical source and the classic (which implies a reversible form of heat transfer), and $dS$ for the increment of the conjugate variable to the temperature $T$. Note that the formula is not an implicit expression for entropy.
In contrast, the mathematical definition for entropy in statistical mechanics is an explicit one:
$$S=k_B \ln\Omega \;,$$
in which $S$ stands for the statistical entropy, $k_B$ is the Bolzmann constant, and $\Omega$ for the number of microstates (or microscopic configurations).
Can these two expressions somehow be connected in a mathematical way? I don't think so (so classical entropy can't be derived from statistical entropy) because if that was the case that would mean incompatible quantities were compared with each other, which by definition can't be done with such quantities (although it is said that $k_B$ forms the bridge between the two approaches to thermodynamics).
Or to put it differently, if there are two different mathematical definitions of entropy, doesn't that mean that there are two different kinds (by which I don't mean different interpretations) of entropy, even if they have the same unit? And can't the same can be said about, for example, the different notions (mathematical definitions) of gravitational force in the Newtonian and Einsteinium light? Or, even more general, between different notions of whatever quantities in whatever two different theoretical approaches of these quantities?
I edited because of some additions where made in the (very clear) answer below. There it's said that the two different approaches are equivalant indeed. But my point is if they are truly equal indeed. The statistical approach is said to be more fundamental from which you deduce the statistical approach, which can only be used in connection with measurements. Or to put it differently, the classical approach is experimental, from which (by first making postulates) a theory can be deduced, while the statistical approach is theoretical which can be tested by experiments. So I don't think that they are the one and only same thing, in concept. Of course, you can say both refer to the one and only true entropy, but I prefer to think that both approaches are (theoretic dependent) referring to truly two different kinds of entropy.
I think it's rather confusing though and nevertheless.
Does anyone have an opinion on this matter? I have the feeling right now though this question belongs more and more in the science-philosophy department and less and less in this one...