In my opinion, a nurse can be both a public health and community health nurse...
For me, a public health nurse is someone who looks at the health of communities. So, TB screenings, infectious disease investigations, mammograms, health education to schools
, etc. to make sure that the community is healthy. In essence, the community is your client and you are trying to prevent illness.
A community health nurse is a nurse who works out in the community (i.e. NOT an institution such as a hospital or nursing home). So, a home health nurse, a HIV case manager, a clinic nurse, etc. However, the person is still your client. So you are still doing direct care tasks such as vital signs, wound care, case management and more tasks that are helping people who are already sick.
To me, that is the difference between public health nursing and community health nursing althogh the two very, very often overlap..
I hope that helps!