Hartree-Fock density is free of self-interaction but lacks electron correlation effects, while the density from KS-DFT (using an xc functional or potential, both which are explicitly density-dependent and gradient-dependent) suffers from self-interaction errors but does incorporate electron correlation effects.
When is the HF density better, and when is the KS density better?
Kieron Burke and co-workers have shown that in many cases one can get better results by using the HF density as input to a DFT-XC evaluation of the energy, as opposed to using the DFT-XC to generate a self-consistent density. This has been termed density-driven error, as opposed to functional-error. SE is especially problematic in cases where an electron is weakly bound, as in many (most) anions. Even small SE in these cases may lead to an unbound electron, as is visible by a positive HOMO energy. Any standard basis set, however, will not allow the electron to escape, and thus produces an energy corresponding to the DFT-XC evaluated on an artificially constrained density. This leads to the energy being sensitive to the size of the basis set, especially including more and more diffuse functions will lead to an energy drift.
I guess this will go in the answer slot, it is a bit long for a comment.
DFT typically has quite a bit less spin contamination than HF (attributed to the inclusion of correlation).
One issue however is that DFT tends to favor electron delocalization, and conveniently HF tends to the opposite. This is part of why hybrid DFT methods get improvements, the over/under electron delocalization is "averaged out". Hybrids also benefit from the HF exchange not suffering from self-interaction, however, range separation methods also correct for self-interaction errors if you don't want to use a hybrid (or apply range-separation to a hybrid for more of a win).
In Hartree-Fock virtual orbitals are akin to electron affinity(determined in the field of N electrons), whereas in DFT they are akin to exciting an electron(determined in a field of N-1 electrons).
Actually saying when HF is better or worse than DFT seems quite difficult to me since you must say what functional you are comparing HF to. There are many ways to fix-up a given DFT functional to overcome deficiencies, two well known corrections being range-separation and dispersion corrections.
To give some sort of summary to what I have said, hopefully others will be more prolific in their answers my instincts say that
I look forward to more answers on this one.
