What is the difference between an American society that is multicultural rather than multiracial?

American society was once described as a melting pot where immigrants came and became American under a multiracial concept. Presently, American society is described as multicultural, where immigrants come to America and retaining their indigenous culture while integrating as part of American society.

What is the difference between an American society that is multicultural rather than multiracial? Are the terms race and culture self-identified, or does a person choose from a predetermined checklist of races or cultures? How has the shift from multiracial to multicultural impacted American society?