Typing on mobile, so will be very brief.

The difference lies in the underpinning of the prejudice. Anti-judaism is generally religiously based, where antisemitism is racially based. In religious anti-judaism, the reasoning stems from the belief that the Jews killed Christ, ergo the Jews are bad. For antisemitism, however, the idea is that the Jews are of a different, inferior race (as compared to other Europeans). Importantly, conversion cannot save a Jew from antisemitism because the perceived fault lies in the blood, not merely on adherence to a particular religion.

Antisemitism, the word, comes from the German Antisemitismus. The creation of the term is usually attributed to Wilhelm Marr in 1879, but similar terms, like anti-semitisch predate it by a bit. Either way, the term was almost certainly coined to both express this difference between the new scientific racism and older religious prejudice as well as to sound more scientific than the term formerly most often used - Judenhass ("Jew hatred").