What is the difference between the state $| \psi \rangle$ of quantum mechanics and the microscopic state $St(q,p)$ of statistical mechanics? In terms of physical quantities, the $|\psi \rangle$ of quantum mechanics and the microstate $St(q,p)$ of statistical mechanics are both a vector, and the microstate of statistical mechanics can be considered as a vector with a size of $3N$, while $| \psi\rangle$ is also a vector, although it's size is unknown.
When the correspondence between the number of microstates and the volume of the phase space is proved by one-dimensional harmonic oscillators to be $\Omega = \dfrac{\Gamma}{\hbar}$ relational proof, the number of energy eigenvalues is regarded as the number of microstates, then whether the number of $\{St\}$ is the same as the number of $\{|\psi\rangle\}$?

$N$ is the number of particles. $\Omega$ is the number of microstates. $\Gamma$ is the volume of accessible area of phase space.
 A: 
What is the difference between the state $| \psi \rangle$ of quantum mechanics and the microscopic state $St(q,p)$ of statistical mechanics?

Apples and oranges.
The $| \psi \rangle$  of quantum mechanics and the $St(q,p)$ of statistical mechanics use mathematics, vectors , tensors and even  complicated functions of special relativity four vectors, but they describe observations through a different mathematical  process.
In physics theories, extra axioms are imposed so as to pick up those mathematical solutions which will have correct units and obey experimentally observed laws. These are called laws in classical mechanics and electrodynamics, and postulates and principles in quantum mechanics; the objective is to have a  theory for physical observations that fits them and, important , is predictive.
It is true that statistical mechanics relies a lot on probability, but the probability postulate of quantum mechanics describes  a probable location for a particle that is biased by the wavefunction, a solution of the appropriate quantum differential equation. The distribution is not statistical, also there is no "number of  $| \psi \rangle$,  $ψ$ is a continuous mathematically function on which creation and annihilation operators act in order for a particle to appear at (x,y,z,t).
