I think your question here exhibits a deeper issue than simply the Fermi-Dirac statistics.Is there a simple explanation for fermi-dirac statistics? I can't understand how a particle can "follow" a statistic.
From that "simple" difference, the statistics arise.https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem said:Bosons are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle: two identical fermions cannot occupy the same state. This rule does not hold for bosons.