Ethnicity=culture shared across generations
Well, the above is a good analytical distinction for intro sociology students, but in practice I do not find the distinction to be all that useful when discussing with any critical subtlety anything related to different categories of people.
We know that racial difference is not strictly genetic difference, since African populations (a.k.a. blacks) actually have more genetic variation than all other human populations combined, and in any case how one divides different populations into different racial categories is a function of culture, not empirically determined biology.
But if race is just superficial physical difference, what categories of physical difference precisely...and who decides? After all, Jews are usually considered "white" in the US these days, but in the past they were not. Ditto with Irish people--and back in the day the nativists swore that there was quantifiable physical difference between Irish and Anglo-Saxons. In any case, what quickly becomes clear if you think about it even for a moment is that so-called "racial difference" is actually used as a discursive proxy for cultural difference. In other words, race becomes synonymous with ethnicity.
If that weren't the case, then it wouldn't even be possible to accuse a black man of "acting white."
Likewise, when Koreans and Japanese consider themselves separate races, are they somehow less right than the Westerners who invariably scoff at the notion?
All human families and communities pass down cultural practices to their offspring, but how we categorize those communities into races and ethnicities as a consequence is--fundamentally--ideological. In short, then, I firmly believe that concepts of race and ethnicity are so thoroughly entangled with each other that trying to make analytical distinctions separates theory too much from practice.