The difference between classical and quantum correlation treatment stems from the statistical nature of describing quantum systems.
Namely, the exchange energy term is missing in classical physics where all mechanical properties are governed by explicit equations of motions and trajectories. In quantum mechanics, this is not possible and the probabilistic nature of wavefunction may bring “new” constraints to quantised energy. The example of exchange energy is for instance explained by the inherent exchange antisymmetry of fermion wavefunctions.