What's the difference between public health and community health?

Subtle: I'd say public health relates to macro health concerns about larger populations and community health is more related to the micro concerns of a specific population or community. You will probably see the terms used somewhat interchangeably depending on the context.

Community vs. PH: In community medicine the primary focus is on individual with emphasis placed on disease diagnosis, treatment, and care for the individual patient. In Public Health the primary focus is on populations with emphasis on disease prevention and health promotion for the entire community.