If I had to speculate, I'd say "Jewish" implies a characteristic: a Jewish banker or a Jewish carpenter is a person who is Jewish among other things. Jew is an identity in itself: you can just say "a Jew", nothing else. And so, if you are an antisemite* and want to suggest that Jewish people are a fundamentally different species rather than a variation within humanity, you use Jew rather than Jewish.
The same pattern appears in racism in general. Racial slurs are usually nouns; you call someone a <slur> rather than a <slur>-ish person.
*or should I say "an antisemitic person"?
