Pohnpei397 has answered this question pretty well. Race strictly has to do with skin color, while ethnicity has to do with culture -- and country of origin. For example, in a group that includes a white South African, a black Frenchman, and a mixed Latina, the adjectives describe the race (skin color), and the nouns pertain to the country and/or area of origin.
Race is based on what you look like, ethnicity is based on what your culture is. That's why you can have Hispanics as an ethnic group that has members of many different races. You can have black Hispanics from places like the Dominican Republic and you can have white Hispanics from anywhere. They're Hispanic by culture and whatever race they are by the way they look.