This is a challenging question to answer, since "the middle ages" spans quite a length of time (roughly 1000 years), and those are pretty big geographical areas. Even if things were uniform, quantifying relative persecution isn't really possible--what's the exchange rate between, say, a pogrom once a century and permanent reduced legal status? I don't think there is one. Also, the answer will by necessity, be far too brief for the topic. I would say that this is the sort of answer you'd need a book for, but looking through my library for sources to answer this question, it's clear that the answer to this would need several books.

Treatment of Jews in Medieval (Christian) Europe was highly variable, both in time and geography. In the early medieval period, Jews tended to have legal status as a community. Jewish courts had jurisdiction within the community, and restrictions were minimal--Jews couldn't own Christian slaves, or couldn't convert them, for instance. And Jewish practice and autonomy was often legally protected, a holdover from Roman law. While not every monarch kept that up, legal treatment was generally positive, and there doesn't seem to have been too much social animosity.

Later on, this shifted. In the 11th century a number of Jews were massacred during the 1st crusade. Blood libels, and accompanying violence occurred in a number of places in Europe. But this wasn't universal in Christian Europe. Quite a few people trying to escape Medieval violence in Central Europe went to Poland, which was (and is) a Catholic country, too (though many from Spain went to Muslim countries, particularly the Ottoman Empire). And Jewish communities still thrived and grew in this area. Legally, Jews were a separate class, generally, directly subordinate to the king. This could afford them governmental protection from hatred, or it could subject them to a king's whims, or to a change of policy with a new king.

In the Muslim world, things were also varied. In many eras treatment was quite positive. The Jewish community of Baghdad was a particularly important one until quite recently, and dwarfed European ones in both size and prestige, both in the local community and among Jews at large. Jews often had a great deal of respect from the government, and diminished legal status didn't mean a whole lot day-to-day in many cases, and was often less negative than in Europe. It was common in the Muslim world for Jews and Muslims to make pilgrimages and worship together, for instance. That doesn't seem particularly likely today, or in Christendom. But the more positive religious views on both ends made this possible. For Jews, Muslims were monotheists with similar laws and customs--for Muslims, Jews were worshipping the right god, and even if they didn't accept his prophet, they were pretty close to the right religion. But for Jews, Christians were polytheists and potential idolaters, and for Christians, Jews were the people who'd killed their god.

But this was not universal. The Almohads, for example, persecuted Jews harshly upon their conquest of Spain, though it was a change for the worse from other Muslims, who'd been friendlier. They also lead a series of massacres in the Maghreb, which brutally attacked a number of Jewish communities. More generally, the tax on non-Muslims was a serious financial obligation for poorer Jews, and failure to pay was a serious legal issue. Jews were legally suboordinate, and generally could not attain positions of power the way Muslims could. Often there were other societal restrictions on Jews, in terms of social actions that reinforced a lesser status of Jews, including distinctive dress. But these were often irregularly enforced, if relations were positive (Spain, at times) or if the Jewish community was remote (rural Yemen).

I know somewhat less about the Jews under Byzantine rule. What I do know is that the precedents of Jewish communal autonomy from the Romans was not so well maintained as in Western Europe, though to what extent varied considerably--Byzantine government couldn't quite decide what Jews were in a legal sense. Jewry-law was a whole category of Byzantine law (much like patent law or rental law is today).

So in sum, I'd say that the average Jew in Christendom had a rougher time with their social place than under Islam. The typical treatment under Christianity (diminished legal status, social restrictions, etc) was the rough treatment under Muslim rule. While expulsions and forcible conversions did happen, they were rare, without the pattern of it in Christendom. But that doesn't mean that every Jew in Muslim areas was better off. A Jew under the Almohad conquest would be in much worse shape than one in Poland during the same time period.

Sources:

• Bachrach, Bernard S. Early Medieval Jewish Policy in Western Europe. Minneapolis: University of Minnesota Press, 1977
• Chazan, Robert. The Jews of Medieval Western Christendom, 1000-1500. Cambridge, UK; New York: Cambridge University Press, 2006.
• Pogonowski, Iwo. Jews in Poland: A Documentary History: The Rise of Jews as a Nation from Congressus Judaicus in Poland to the Knesset in Israel. New York: Hippocrene Books, 1993.
• Hecht, Neil S., ed. An Introduction to the History and Sources of Jewish Law. Oxford : New York: Clarendon Press ; Oxford University Press, 1996. Publication / the Institute of Jewish Law, Boston University School of Law no. 22.
• Frassetto, Michael, ed. Christian Attitudes toward the Jews in the Middle Ages: A Casebook. New York: Routledge, 2007. Print. Routledge Medieval Casebooks.
• Chouraqui, André. Between East and West a History of the Jews of North Africa. Philadelphia: Jewish Publication Society of America, 1968.
• Cohen, Mark R. Poverty and Charity in the Jewish Community of Medieval Egypt. Princeton: Princeton University Press, 2005.
• Lewis, Bernard. The Jews of Islam. Princeton, N.J.: Princeton University Press, 1984. Print.
• Masters, Bruce Alan. Christians and Jews in the Ottoman Arab World the Roots of Sectarianism. New York: Cambridge University Press, 2001.
• Meri, Josef W. The Cult of Saints among Muslims and Jews in Medieval Syria. Oxford; New York: Oxford University Press, 2002. Print.
• Stroumsa, Guy G et al. Jews in Byzantium: Dialectics of Minority and Majority Cultures. Leiden; Boston: Brill, 2012.
• Tobi, Joseph. The Jews of Yemen: Studies in Their History and Culture. Leiden; Boston: Brill, 1999.