Off the top of my head the two primary justifications for an HRG system are:
1.) Safety. Arc flash is not as big a concern when you have a phase to ground fault relative to a solidly grounded system.
2.) Orderly fault recovery. Systems are designed to alarm in the event of a ground fault but remain energized for a specified period, e.g. 24 hrs., before opening a protection device. This would allow orderly coordination of corrective action.
A significant thing to be aware of is that the grounding system design (or lack thereof) must be taken in to consideration when setting up devices such as VFD's.
Be aware that some systems such as the brand I-Gard provide trouble shooting tools to help pinpoint a fault.
There is much debate about the merits of an HRG system. As a guy who is around the field equipment I welcome the reduced arc flash hazard.
I work in a facility with a mix of transformer secondaries and grounding. Grounded delta (intentional), grounded delta (unintentional), solidly grounded wye, ungrounded delta, and a new HRG system. Systems and facilities as new as two years old, some that date back to the turn of the century, and I ain't talking about the most recent one. Think wooden beams and wooden pegs...
To date, we have had no problems with the HRG outside of some VFD config confusion, and a case where we ignored the ground fault alarm and suffered an unplanned uncoordinated shutdown 24 hrs. later. I am aware of at least one case where we had a workmanship problem that caused a ground fault at a conveyor motor where we were able to conduct an orderly shutdown and resolve the problem with minimal impact. I know there have been other similar cases but I have not been directly involved.
There is some interesting reading, both pro and con, on the internet. Search for the I-Gard system.