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
- HF would possibly be better for electron affinity when using a frozen orbital assumption (assuming a low quality bare-bones DFT functional).
- DFT provides a better density for systems where spin-contamination is an issue.
I look forward to more answers on this one.