If someone discriminates against white people they're racist, if someone discriminates against Christians they're just seen as prejudiced against that group.
But I can't help but notice that there are entire sets of words that are regularly used to refer to prejudice against the Jews such as antisemitism or xenophobia. Hate-speech against Jews would be called "antisemitic hate speech" while hate-speech against any other race or religion would probably just be called "hate speech."
Is there a specific reason for this?
If you want to know the origin of the word "anti-Semitism", that's a fair question. But if you want to simply know what words are used to describe anti-Jewish hatred compared to other hatreds, you would be incorrect to say anti-Semitism is a unique word.
But I can't help but notice that there are entire sets of words that are regularly used to refer to prejudice against the Jews such as antisemitism or xenophobia
Xenophobia is not used to refer only to Jews. It refers to many other groups. Xenophobia and jingoism are terms that were used in US rhetoric throughout the 20th century, particularly in relation to immigration policy and foreign policy.
Hate-speech against Jews would be called "antisemitic hate speech" while hate-speech against any other race or religion would probably just be called "hate speech."
There are other groups that also receive similar terms for hate against them. For example, the terms "Islamophobia" and "homophobia" and "transphobia", all apply to specific groups. For how the terms developed, the question might be suited to /r/asklinguistics.
There is one term for this, anti-semitic. Xenophobic means fear of outsiders/foreigners. Other groups which commonly experience discrimination have their own term too- misogynist comes to mind.