The simple answer:
Judaism is a religion, mbodied by following the laws given to Moses, by God, at Mount SInai 3500 years ago, the laws being embodied in the FIve books of Moses (Torah) and the Mishnah (Oral Law).
Zionism is an ideaology which is most simply defined as Jewish Nationalism. When many people speak of Zionism they refer to the political movement founded by Theodore herzl in the nineteenth century.
The state of Israel is a modern day state, founded by UN declaration in 1948. It is controlled by a government, a system of laws and a court system. Who controls the state of Israel? The people who elect the government democratically in democratic elections. There are parties representing all the countries ethnicities and groups, including Christians and Arabs.
As such, many people within Israel may be Jewish, and many others may be zionists, but neither can be said to control thhe country except to the extent that Jews and Zionists are the majority and thus have the largest representations in te countries parliament (The Knesset)