Anti-Semitism is hostility, prejudice, or a hatred towards Jews. Anti-Zionism, on the other hand, is an opposition to Zionism. Zionism is best defined as a political and nationalistic movement of Jews and Jewish culture that aims for the goal of re-establishing a Jewish homeland in the historic land of Israel. The idea behind Zionism was, and still is, that Jews, like any other group of people, are entitled to a homeland. Theodor Herzl is considered the founder of the modern-day Zionist movement, and hoped to build a Jewish homeland in the twentieth century. Interestingly enough, there was Jewish opposition to building such a homeland. Secular Jews wanted to assimilate and cast away their Jewish identities. There were also a sizable number of religious Jews who thought it was heretical since they were operating under the idea that only with the coming of the Messiah can there be a Jewish homeland in Israel once more. However, the Holocaust changed all of that, and Zionism became much more popular among Jews post-WWII, which is most notably illustrated by the creation of the Jewish State in 1948. With that, anti-Zionism is the opposition of a Jewish homeland, although some have attempted to inaccurately narrow the definition to mean "having a problem with Israel's policies."
Anti-semitism before the creation of the State of Israel was much easier to point out. During the Middle Ages, Jews had to contend with anti-semitism of Christian and Islamic varieties. Christians tormented Jews through segregation via ghettos, pogroms, expulsions, and executions. Muslims relegated Jews to a second-class status known as dhimmi (ذمي). In the 19th and 20th centuries, the hatred towards Jews was racial. The question here is whether anti-Semitism simply evolved on a global level through the guise of anti-Zionism.
The problem with equating anti-Zionism with anti-Semitism is that one can, at least theoretically (if not more so), have a problem with the creation of the State of Israel and still be a philo-Semite. There are some ultra-Orthodox Jews (e.g., Neturei Katra) and liberal Jews who have a problem with Zionism without necessarily succumbing to Judeophobia. There are also some who believe that Zion was meant to be more of an idea than it was meant to be a nation-state. While there is both a theoretical distinction, as well as at least some differentiation in practice, the sad truth is that the confluence between anti-Zionism and anti-Semitism is so considerable that trying to differentiate between the two these days is like trying to split a hair with a knife.
Although Zionism is about the existence of a Jewish state in the historic land of Israel, some like to contort the definition of anti-Zionist into meaning "critical of Israel." For argument's sake, let's briefly assume that that definition is correct (and I'm sure self-labeling anti-Zionists prefer this definition because it's more palatable). Criticizing Israel is not anti-Semitic unto itself. Israel, like any other government, is run by fallible human beings. There are bound to be mistakes made in crafting and executing public policy. Criticizing the Israeli government is practically a national pastime among Israelis. Three Jews, five opinions, am I right? But in all sincerity, how do we distinguish between legitimate criticism of the state of Israel versus anti-Zionism merely acting as a front for anti-Semitic behavior?
If a critic consistently singles out Israel while ignoring far worse crimes against humanity, that is anti-Semitism. If someone likens Israel to Nazi Germany or invokes traditionally anti-Jewish stereotypes or depictions (see here), you're dealing with an anti-Semite. If one is attacking the merits of Israel's right to exist instead of merely criticizing the Israeli government's policies, odds are that you're dealing with an anti-Semite. Whether we like it or not, many criticisms of Israel has morphed into something well beyond legitimate concerns with how the Israeli government behaves.
Look at how the United Nations puts disproportionate emphasis on the Israeli government while ignoring or downplaying much greater atrocities against human rights. Look at how hateful the BDS movement is in its attempts to cripple Israel. Look at the slanderous claims of apartheid being made against Israel, whether we're talking about what goes on in Israel proper or in the West Bank. Look at how anti-Semitism (especially in Europe) hasn't reached such high levels since right before WWII, and how much of that anti-Semitism is linked to anti-Zionistic sentiments. Look at this recent Washington Post article from Larry Summers as but another piece of evidence of how virulent anti-Semitism (much of it conflated with sentiments regarding Israel) is on American college campuses, even in spite of the hypersensitivity towards racial prejudice and other politically correct causes célèbres. Anti-Zionism has devolved into being more blatantly anti-Semitic, and it is tantamount to saying that Jews have to live beneath, between, or among other people, that they don't have a right to be their own nation, that they should remain subordinate as minorities. Sadly enough, hating on Israel is probably one of the only things that the far Left, the far Right, and militant Muslims can all agree on.
Singling out Jewish self-determination for condemnation is, by definition, discriminatory and bigoted, although don't tell anti-Zionists that they are racist because they will simply double down in their animus. Given how much the State of Israel has played into Jewish identity politics, it should be no surprise that the line between anti-Zionism and anti-Semitism has become thinner and thinner since the creation of the State of Israel. Israel has become a nation-state that has thrived in spite of animosity from its neighbors in the Middle East. As Israel becomes more successful and build more relations with other nations (e.g., India, Russia, China), I can only expect anti-Zionism to become more anti-Semitic, at least until these anti-Semites can see the error of their ways.