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.