It's more complex than that, but it's damned hard to explain why. One part is the hereditary issue, but there's also that thing about Jews who identify as such and are atheists or totally non-practising. And although our culture and history is spiritualized, our secular culture and history is "consumed" (for lack of a better term) in the way American cultural history or German cultural history might, especially in the modern period. Ugh, I wish there was a better way of grasping at what separates Judaism from Jewishness and I'll keep trying, but the best I can do right now is to tell you to watch My Big Fat Greek Wedding. I know that sounds lame and unscientific (in terms of its imprecision and the reliance on a movie of all things), but just do it. You learn a lot about how Jewish culture and Jewishness works by looking at real-world and artificial representations of other Mediterranean-based cultures.
Edit: Just to clarify, I'm not saying that MBFGW is supposed to explain to you what it means to be a Jew. I mean that the way in which it goes about explaining what it means to be Greek is similar to how I might explain what it means to be a Jew; the movie itself makes plain that it's more than a racial, national, or religious thing, but rather it encompasses all aspects of the lives of those who retain that identity, in ways that it's difficult to explain in words.