This is an interesting question - I'm a student of Jewish studies and art history so I hope I can help.

There is a continuous debate on where to draw the line between anti-Judaism and antisemitism. The persecution of Jews in the Middle Ages, however, stemmed from differences of religion. Scholar Jacob Rader Marcus compiled a great (and timely) book in 1938 called "The Jew in the Medieval World", which can be found in full here, a source book of documents from the Medieval era relating to Jewish life. The majority of my answer will be based on the texts included in this book, and I will link to other sources where appropriate.

In his introduction to the text, Marcus emphasizes that anti-Jewish sentiment of the Middle Ages stems originally from the reduction of Jews to the status of second-class citizens under Constantine the Great (spurred by his conversion to Christianity and the establishment of the Roman Empire as a state of "Christian religious totalitarianism"), a theme which continues through the Medieval era. Anti-Judaism of course existed prior to this in Greek and Roman societies, but Christian Roman laws (for example, forbidding the treatment of Jewish patients, burning Jews who commit crimes against those who have left their faith, and outlawing the building of new synagogues) helped set the stage for Medieval persecution of Jews.

We can see this continuation in examples such as Medieval Spain, where a large number of Jews had settled and witnessed their culture flourish under the Abbasid Caliphate. However, the newly-enacted Pact of Omar (text can be found in Section I, chapter 3 of Marcus' The Jew in the Medieval World) heavily taxed Jews, made them differentiate themselves from Muslims through style of dress, and again forbade them from erecting new synagogues (among other things). When Christians retook Muslim Spain during the Reconquista, Spanish Jewry again came under threat due to differences in religion. The Spanish Inquisition regularly led campaigns against Jews accused of heresy, burning and banning the Torah, burning Jews in the streets, and exiling European Jews. This was not an uncommon sight throughout the Medieval era -- Jews were expelled from France in 1182 (document #5 in Marcus' book), the Black Death was often attributed to Jews (document #9), the Talmud was burnt in Paris (document 29), Jews were expelled from Spain in 1492 (document #11), Jews who had just converted to Christianity were massacred in Lisbon in 1506 (document #11), and so forth, all due to religious conflict between Christianity and Judaism.

Jews that survived persecution under the Spanish Inquisition fled to places like Italy, where religious differences again played a large role, making Jews a financial asset due to the system of mercantilism and fostering hatred between Christians and Jews as Christians were now dependent on and indebted to Jews due to the fact that they were able to handle money in ways that Christianity forbade. The first ghetto was established in Venice and some stereotypes about Jews that are still prevalent (i.e. that they're greedy and money hungry) stem from their time in Italy, and this is a religion-based stereotype rather than an ethnic one. In nearby Germany, accusations of blood libel (again, emphasizing religious differences) were not uncommon, and were even depicted in art (like in 1475's Martyrdom of St. Simon of Trent). Martin Luther accused Jews of being Satanic beings in writings such as "Concerning the Jews and their Lies" (included in Marcus' book alongside other writings, Section II chapter 33).

In terms of art, this article by Sara Lipton provides a great overview. Throughout the Middle Ages, Jews were defined in art by their pointed hats (which Lipton attests they never actually wore) rather than their supposed physical characteristics until late into the 12th century. Lipton states that no Christian texts actually stated that Jews had different physical characteristics (specifically the "Jewish nose") but that this caricature came about in response to a new trend of depicting divine suffering in art and that the emphasis on the nose was actually to emphasize the angle of the depicted Jew's head looking away from Christ. She notes that there was not one single "Jewish nose" depicted, but the later shift into artistic "realism" and the 13th century study of physiognomy consolidated these features into the "Jewish nose" stereotype. I cannot personally vouch for the validity of these claims (I focus on early Soviet art rather than Medieval art), but it is clear that well beyond the 13th century's apparent focus on facial features and physiognomy, Jews still were not consistently depicted in art as a hook-nosed caricature. If you're able to find higher-res images of Johannes Pfefferkorn's series of religious pamphlets from the first decade of the 1500s (eg. "Speculum adhortationis iudaice ad Christum", "The Confession of the Jews", and "How the blind Jews celebrate their Easter"), you will notice that rather than being reduced to what we may see as an ethnic caricature, Jews are portrayed as more or less average looking people.

I hope this helps with the Medieval part of your question!

EDIT: TL;DR the persecution of Jews in the Middle Ages wasn't due to an ethnic difference as even just differentiating between what a Jew and a Gentile stereotypically (physically) looked like wasn't being fleshed out until the 13th century (and even then it wasn't extremely common). Jews were, however, persecuted for "heresy" and religious differences pretty much constantly, and we've even gotten some nasty longstanding stereotypes out of it!

I'd like to address a different angle to your question- specifically the idea of Jews as a race as opposed to a religion.

There is a very good case to state that the era of the Spanish Inquisition innovated the idea of Judaism as a race, as well as racial- not religious- differences between peoples. I wrote about this a while ago (see the second comment)- before this time, willing (not forced) converts to Christianity were treated as Christians like any others, but afterward they were considered part of a different "blood line" and through that were considered "lesser" and without "blood purity" (limpieza de sangre). (For a good overview on this, see Jane Gerber, The Jews of Spain.)

So basically, until the Inquisition's era, Jews were considered bad, and depicted as having bad traits (though they were probably more likely to be depicted as having devil horns and a monster's body than just as regular people with weird features), because they were Jewish- but once they converted they could be saved and become just like any other Christian. Not so afterward.

Once you bring it up, scientific racism came much later. I highly recommend the (unfortunately named) book Defending the Master Race by Jonathan Spiro, which discusses the many elements of race theory, anthropology, eugenics, etc. that came together to become scientific racism, specifically in the form of Madison Grant's book The Passing of the Great Race. It came later because many of the real scientific discoveries in the fields of anthropology, heredity and genetics which led to the twisted versions in race theory and eugenics were only discovered in the 1800s.