I think the difference is in quantum many body problems we usualy are dealing with indistinguible objects, with fermions for example the wavefunction must obey the pauli exluction principle, this means that when u solve the system for every object solo, u not only miss corrrelation but also exchange term coming from objects being indistinguible. While the classic analog there is no need to have all permutions cause U can tell the object apart, so that there is no exhange. This means that u can calculate your system for every object seperatly add the energy and the difference between the energy of the total system and this split calculation gives u the correlation energy.