Race and ethnicity, on the one hand, and culture on the other affect us in different ways. In general, race and ethnicity impact us because they affect the ways in which other people perceive us. (Race and ethnicity can also affect us by affecting the culture in which we grow up.) Culture affects us because it affects the ways in which we ourselves behave.
In the United States, race and ethnicity tend to affect how other people see us. People tend to think of whites as the norm in the United States and to have stereotypes about non-whites. Asians, particularly East Asians, are seen as high achievers while African Americans and Hispanics are generally perceived in a less positive light. Non-whites have to live with these stereotypes. At times, they even have to deal with more overt acts of racism or prejudice.
While race and ethnicity affect how other people see us and treat us, culture affects how we act and think. (And remember, race and ethnicity can affect our culture.) If we grow up in a rural, conservative area, we will tend to have a different culture than if we grow up in an urban area. People from the South might have a different culture than people from New York or Boston. Our culture affects how we think about many things including religion, patriotism, gender roles and others. It affects the kinds of things we like to eat, the clothes we wear and the music we listen to. In these ways, race and ethnicity on the one hand and culture on the other both affect us, but in somewhat different ways.