Maybe I could give a legal distinction.
Anti-Semitism is a set of human rights violations, a wide variety of forms of discrimination against the Jewish community. Anti-Zionism is the rejection of one particular Jewish right, the right to self-determination of peoples.
There is a difference between anti-Zionism and being anti-Israel. Let's say you hate the way the roads are paved in Israel, and so you're critical of Israel, that's not being anti-Zionist. Anti-Zionism goes to the existence of the state of Israel rather than to its behaviour.