What is the difference between Electron Exchange and Electron Correlation and why do these have to be separately accounted for in DFT over and above Columbic repulsion energy?

Electron correlation, refers to all the effects that are missed when the electrons are treated as independent.  I would elaborate but I think Wikipedia is a little more eloquent than I am (Electronic correlation):

The word correlation energy has to be used with caution. First it is usually defined as the energy difference of a correlated method relative to the Hartree–Fock energy. But this is not the full correlation energy because some correlation is already included in HF. Secondly the correlation energy is highly dependent on the basis set used. The "exact" energy is the energy with full correlation and full basis set.

In DFT the other electrons are treated as a mean field and because of this the interaction between electrons isn't captured exactly.

Electron exchange referes to the Pauli Exclusion Force which arises from the fact that the system has to satisfy the Pauli Exclusion Principle.  In essence no two electrons with the same spin can exist in the same orbital.  This is a quantum effect that is not related to classical electrostatic repulsion.  Attempts are made at including these effects (see Density functional theory), however at the end of the day these are all just approximations.

Reference: https://www.quora.com/What-is-the-difference-between-Electron-Exchange-and-Electron-Correlation-and-why-do-these-have-to-be-separately-accounted-for-in-DFT-over-and-above-Columbic-repulsion-energy