What is the difference between statistical mechanics and quantum mechanics in terms of accuracy?
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.