The difference is in how they are paid.
"Public health" implies public payment, in the form of grants, donations, sliding scale fees, tax dollars, etc. Your county health department, local parish nursing programs
, and school nurses are examples of public health services.
"Community health" implies private or third party payment for out-of hospital services, including insurance, Medicare and Medicaid.
Prior to the existence of Medicare, virtually all out-of-hospital services were financed publically. Most Visiting Nurse services were funded solely by public money prior to the advent of Medicare. They started billing for services when Medicare came into existence.