Identical quantum particles fall into two fundamentally different classes. Bosons have many-particle states that are symmetric under particle exchange; any number can occupy the same one-particle state. Fermions have antisymmetric states, so no two identical fermions can occupy the same one-particle quantum state.

For noninteracting particles in thermal equilibrium, the mean occupation of a one-particle state of energy $E$ is $$\bar n_{\rm BE}(E)=\frac{1}{e^{\beta(E-\mu)}-1}$$ for bosons and $$\bar n_{\rm FD}(E)=\frac{1}{e^{\beta(E-\mu)}+1}$$ for fermions.

The sign difference has major physical consequences. For fermions, $0\le\bar n\le1$ for each state, producing Fermi surfaces, degeneracy pressure and the electronic structure of matter. Bosonic occupation can become macroscopic in low-energy states, enabling phenomena such as Bose-Einstein condensation.

When $e^{\beta(E-\mu)}\gg1$, both distributions reduce to the classical Maxwell-Boltzmann form $$\bar n\approx e^{-\beta(E-\mu)}.$$ Thus classical particle statistics emerges when occupation of each quantum state is sparse.

Electrons, protons and neutrons are fermions; photons and many collective excitations such as phonons are bosons. Quantum statistics is therefore not a small correction to classical thermodynamics: it controls the structure of atoms, solids, compact stars and low-temperature quantum matter.