Can someone explain to me...

A private duty nurse is hired directly by a family or individual needing frequent or daily nursing care in their home. A home health nurse usually works for an agency and provides nursing care in their clients' homes. A private duty nurse (I think) has only one client at a time, whereas home health nurses have many patients at any given time. I'm no expert, so I may be incorrect with some finer points here, but that's my impression of both jobs.

Private duty is one-on-one care done in shifts at one home, home health is broken up into visits, lasting 30 minutes to an hour with the nurse visiting several homes in a day.

And you can be a private duty nurse through an agency as well as hired directly by the family.

was a time when private duty nurses also worked in hsp...paid by the pts family to give one on one care. Hsp had regestries.

I've been both. There was a time when a family could hire and pay for an individual nurse, LPN or RN, to take care of their family member one-to-one. Sometimes it was thru an in-hospital registry, sometimes an agency. The last time I did this was 15 years ago! And sometimes I had to tell the family that mom or dad did not really need that level of care. I learned that some fortunate people just wanted 'company' for their family, and were willing to pay for it.
Private duty is certainly done in the home as well; usually thru an agency, and most often for high-tech care, such as patients on vents. This is sometimes referred to as in-home shift work. Done that, too, and it can be very taxing.
Traditional home-health is frequently called 'visiting nurse'. A short visit, 30-60 min usually, is done at the pt's home. Dressing changes, new diabetic teaching, post-open-heart surgery care, weekly blood draws. These are done when pre-approved by insurance, medicare, medicaid. This is my favorite type of nursing!