I sort of disagree with iain
, but this is no big deal
, since where there are two Jews, there are three opinions
I'd say that Jewishness is an ethnicity, and that Judaism is a religion. The two overlap, but are not identical. Members of Beta Israel identify themselves and are recognized as members of either or both categories.
As for the concept of race, I don't see it as relevant to the nature of Jewishness, since the only racial category that has any precise meaning is "the human race."