One is anti-Jewish, one is anti-Israeli policy
Anti-Semitism is anti-Jewish bigotry and unacceptable. But anti-Zionism is simply being against Israeli policy and a legitimate political belief.
Arguments against anti-Zionism are rooted in international law and human rights. Anti-Semitic arguments are rooted in bigotry and hate.
The root of anti-Semitic and anti-Zionist arguments also highlight the differences between the concepts and demonstrate that one is bigoted while the other is political opposition to Israeli policy.
Anti-Zionist arguments are rooted in International Law and the Universal Declaration of Human Rights. Many anti-Zionists oppose Israel's occupation of Palestinian lands on the grounds that it is illegal and nothing more. Anti-Zionist belief is rooted in equality and upholding the basic human rights of the Palestinian people. Anti-Zionist arguments are not borne from racial prejudices or resentment like anti-Semitism is. 
Not all anti-Zionist arguments stem from international law and the respect for human rights. Some subscribe to anti-Zionist beliefs for religious reasons. For example, the American Presbyterian Church holds anti-Zionist beliefs not because of a respect for Palestinian human rights, but because it believes Jews "are condemned to suffer for their rejection of Jesus and complicity in his death".  These beliefs are clearly anti-Semitic. Consequently, those who hold anti-Zionist beliefs based on purely religious beliefs are expressing anti-Semitism in their anti-Zionist beliefs. This means that in some instances, anti-Zionism is equivalent to anti-Semitism and the two have a much closer relationship with some aligned goals.
[P1] Anti-Zionism is not rooted in bigotry. It is borne out of a respect for international law and human rights. [P2] Anti-Semitism is borne from prejudice and bigotry. [P3] Therefore, anti-Zionism is acceptable and distinct from anti-Semitism and anti-Semitism is bigotted and prejudiced and socially unacceptable.
[Rejecting P1] Anti-Zionism can also be rooted in bigotry and prejudice.