In my sociology class I learned that cultural assimilation is when individuals from one culture adopt aspects of a new dominant culture. (religion/language/norms) For example, a Japanese man new to the US must culturally assimilate himself in society by learning English.
Structural assimilation is when an ethnic group is fully incorporated into a society--they have equal access to institutions (government, education, etc). For example, African American's holding positions of authority in the US means that they are structurally assimilated.