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It is my impression that everything relevant to 'quantum' is the truth and others is just a simplification. Is this the case in those two kinds of mechanics?

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Quantum mechanics is about the physics of very small things, molecules and smaller.

Classical mechanics is about macroscopic things. Quantum mechanics covers the whole of classical mechanics as well, but in the macroscopic limit both become equivalent. For example discretized energy states become so close, that you can thing of them as a continuum of states.

Statistical mechanics can be employed to consider many, quantum or classical, like systems and how they evolve. One application is to get thermodynamic quantities from quantum mechanical energy states, by considering how these states are statistically populated.

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From plain Wikipedia,

Statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system, where the state of the system is uncertain.

(Thus, very useful for things like thermodynamics, where everything we measure is an average in some sense.)

Statistical mechanics per se doesn't rely on any specific set of laws for the interactions of the "small parts" that compose a mechanical system. This set of laws could come from either quantum or classical mechanics.

So, in a way, the comparison posed is not valid. But, by comparing the statistical treatment that uses classical mechanics with the one that relies on quantum mechanics, the later gives us a broader agreement with experiment.

Again, from plain Wikipedia,

The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. (...) Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.

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They operate in different ways, quantum mechanics allows us to derive, for example, the energy levels of molecules and statistical mechanics allows us to study the effects produced when these energy levels are populated, usually thermally via the Boltzmann distribution.

Neither theory has been shown to be false under the conditions where they might be expected to apply. However, in quantum mechanics there is no satisfactory derivation of the Schrodinger equation, he just wrote it down. Therefore, even though QM works for all molecules and explains very many phenomena there remains the possibility that it will eventually be shown to need modifying.

(note : While Qm is often thought to only apply to small things, the quintessentially quantum Double-Slit experiment, often performed with photons and electrons, has now been performed on molecules as large as $\ce{C60}$. So there is no clear distinction between 'small' and 'large'. Similarly Bose-Einstein condensates can contain many thousands of atoms. Remarkably, White dwarf stars are thought to be supported by 'electron degeneracy pressure', a consequence of the Pauli exclusion principle.)

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