How is the anthropological concept of culture distinct from the Western concept of race?
The best way to answer this question is to emphasize the word "Western." In other words, a Western concept of race has all the blind spots, cultural biases, and commonsense thought patterns of the West. So, it is not objective truth, but a Western understanding of a topic - in this case, race.
Anthropology is the study of people and culture. If it is done well, then it is self-critical and honest about its own presuppositions and assumptions. Moreover, it seeks to be neutral as possible. If this is done, then the anthropology will be much more objective.
With that background, we can say that race and culture are two different concepts in their own right. To make a general statement, race in the West is usually based on external features; in a word, how a person looks. Culture from an anthropological point of view has very little to do with race. It seeks to understand how society works.
The concepts of culture and race are distinct because one is seen as a physical characteristic while the other is seen as the sum of everything that is not physically transmitted from one generation to the next.
The Western concept of race is focused on the physical appearance of the people involved. Race, therefore, is a physical characteristic. By contrast, culture has to do with things like styles of dress, kinds of food, traditions about marriage, and everything else that is not genetic.
Therefore, race and culture are very distinct concepts. People of the same "race" can have different cultures (think about Americans as opposed to Russians, for example) or people of different races can share a culture (Americans of different races, for example).