The two are always conflated by the extreme ultra-right.
Zionism - the late 19th and early 20th century movement, led primarily by Jews, to create a Jewish ethnostate where Israel now exists. Based upon the myths found within what Christians call the Old Testament, and the Jews the Tanakh. They believe God gave them this land, and that therefore anything they do (or did) to those already living there was (and is) perfectly justified.
Anti-semitism - hatred of those who are classified as Semitic, i. e. Jews AND Arabs.