In OTL, the Roman emperor Vespasian, fresh off his victory over the Jewish rebels of the Roman-Jewish War of 66-73 CE, imposed a special tax on all Jews as a punishment for their revolt. The tax went to the Roman Temple of Capitoline Jupiter, perhaps the holiest site in ancient Roman paganism, and was a substitute for the older tax that the Jews had to pay for the Temple in Jerusalem.
Also, in OTL, a later emperor, Nerva, exempted early Christians from the tax. Christians now had a financial advantage over Jews, in spite of persecutions every now and then.
The Emperor Constantine would, one day, centuries in the future, convert to Christianity, and forever alter the balance of religion in ancient Rome.
What if it had been a liberal version of Judaism instead? What if it had been, say, something like Reform Judaism, and it had been backed up by more conservative Jews as a missionary activity?
Vespasian goes easier on the Jews in ATL 73 CE and does not impose the tax. There is thus no reason for Nerva to exempt Christians from anything.
Christianity is seen by many Romans as a weird branch of Judaism, and, with a liberal version of Judaism to turn to, the Romans go for the latter, which strikes them as less "weird" and more "rational".
In 303 CE, Emperor Diocletan bans Judaism (as Christianity is too small a group to be worth making a fuss about), and he orders the synagogues burned, along with Torah scrolls and other Jewish paraphernalia. He orders Jews to be thrown to the lions, tortured, murdered, and otherwise harassed, and this persecution continues until 311, when Emperors Galerius, Constantine, and Licinius sign off on an "edict of toleration".
In 312, while fighting the Battle of Milvian against Maxentius, Constantine sees a blazing menorah, or a burning bush (the accounts differ from historian to historian), and is told, "in this sign and in the name of My mitzvot, you shall conquer". The menorah is inscribed on Constantine's soldiers' suit, and Constantine eventually converts to Judaism.
In 313 CE, Constantine reaffirms that the Jews are an accepted religion in the Roman Empire, However, he tries to meddle in the religion (much like he meddled in OTL Christianity), trying to uphold orthodoxy and doctrine. However, this fails, for two reasons: 1) Judaism is more of an orthopraxy (correct-action) than orthodoxy (correct-belief) religion, and 2) Judaism is harder to govern, as it's not quite as "cultic". Reformers send missionaries out to the barbarians in order to spread their brand of Judaism, thus circumventing the "orthodox" Jews in Constantine's Sanhedrin.
While Constantine mulls putting together funds for a Jewish Temple, he is a bit stubborn in preferring Rome to Jerusalem. This controversy continues through subsequent emperors, and they quietly bury the notion of funds for a Third Temple under other, allegedly more important, funding, such as the improvement of the Roman military and the beautification of the Eternal City.
Julian the Apostate
Julian is a pagan, and he is the last of the pagan Emperors (just as in OTL). In ATL, however, he doesn't commit any funds to the reconstruction of the Jerusalem Temple, as Judaism is the religion he is protesting against, rather than Christianity. He starts funneling funds towards Christian churches, the Jovian cult, the temples of Aphrodite/Venus and Priapus, and various other pagan cults and religions. He dies after a battle against the Persians. The next emperor restores Judaism's high position among the Imperial religions.
In 400 CE
The Sanhedrin of Rome, composed of rabbis and priests, come together under Emperor Honorius to continue the work begun with the "Mishnah", the codification of the Jewish "oral law". (The "Reformists" reject this "oral law" as an unbiblical addition to Judaism that they don't need to deal with.)
The result is the Roman Talmud, which is more complete (and more Imperially and classically oriented) than the OTL Jerusalem Talmud would have been. It is more comparable to OTL's Babylonian Talmud, except with a heavier Greek flavor. (One might think of Maimonides' OTL work "Guide for the Perplexed" and its appeals to logic and reason.)
In 476 CE
Odoacer marches into Rome and takes it over, ending the Western Roman Empire. He is a German Jewish reformist who takes a hands-off attitude to the Roman Sanhedrin and its workings. Unfortunately, by this point, the Eastern Roman (Byzantine) Empire has adopted Sadduceeism (Temple-priestly Judaism), and dislikes the Talmudic Pharisaism that the West advocates. A schism is underway, and it is a major one. Aside from the Reform wing, there are some Karaite and Samaritan "heresies", and new groups are often forming. The result is that Italy and Byzantium are always getting on each other's nerves.
Early Middle Ages
Judaism, instead of slipping off to Babylonia (Iraq) as it did in OTL, remains primarily a Mediterranean and European religion. Its flavor becomes more like that of OTL Islam, with people willing to fight for their beliefs and to impose those beliefs, an equivalent of a Caliphate (a Rabbinate), and multiple bickering schools of jurisprudence. Instead of OTL Sura and Pumbeditha, we have ATL Paris and Rome and Constantinople, and nascent schools forming in Venice and Ravenna. Some schools are more legalistic than others, and just about everybody (in the more orthodox camp, anyhow) looks up to Rome.
Nonetheless, in spite of the change of religion, a quarreling rabble of barbarians is still a quarreling rabble of barbarians. The Huns, Vandals, Goth, Italians, and Byzantines still fight each other, civilization still declines, and anarchy triumphs briefly in some quarters. Through this mess, Jewish missionaries wander around Europe, spreading the wisdom of Hillel (or Moses, or Josephus the historian, or some other inspired rabbi or priest), along with the Torah. A brave few revert to the paganism that preceded Judaism (but it's Canaanite Paganism, not Roman Paganism) - yet another "heresy".
In time, the Franks arise, as do the Germans, the English, the Spaniards (Visigoths), and some other familiar nationalities. However, their ancient temples get replaced by synagogues, rather than churches, and matzah is made everywhere come Passover time.
The Byzantines rebuild the Temple in Jerusalem by 500 CE, and they thus mark themselves as the "center of the world". Emperor Aristotle Benjamin moves the capital of the Empire from Constantinople to Jerusalem in 510 CE, thus restoring the ancient capital of David and Solomon. He anoints a High Priest and reinstitutes the Temple sacrifices, which the western Rabbis disavow. Politics in the European world now revolve around its eastern end, where the wealth and the economic power comes from.
622 CE or thereabouts: Muhammad
Muhammad the Imposter or Muhammad the Heretic, as official Byzantine sources later call him, arises in Arabia, professing his own brand of monotheism. However, it faces stiff competition from established branches of Judaism, and it is dismissed by many as a knockoff of the original religion. Muhammad gathers a devoted following of Arabs, however, and they give rise to one of the Arabian peninsula's more colorful cults, that of Islam.
Due to butterflies, the country of Burgundy persists into the modern era, instead of being absorbed into France and Germany. It is unclear why this happens, but religious disputes may have something to do with it.
11th century to mid 15th century CE: Genghis Khan and the Mongols
Genghis Khan and his successors come off of the Asian steppe at about the same time as he did in OTL (ATL Jews fight rather like OTL Muslims and some OTL Christians), and he overruns many of the same cities. Jerusalem is well-armed and well-defended, however, and the Mongols find an invasion of the Levant harder to accomplish than in OTL, so they stay more to the north, penetrating past the Rhine into Burgundy and France, with Paris narrowly avoiding a sack in 1256. Eastern Europe's population has taken a steep fall, which is made worse when the Black Death rolls around in the mid 14th century.
At least, in ATL, there is no rumor-spreading about Jews poisoning wells and taking the blood of innocent Christian children. Rather, the rumor-mongering is about "Baalists" and Thor worshipers who bow to "the head of a donkey" and slaughter innocent Jewish children, as well as slipping poison into the waterways. Like the OTL blood libel, those rumors are baseless, but widely believed. Rioters gather in the streets, tie suspected pagans to wooden posts, and pelt them with rocks until they expire.
Rabbis issue warnings about "loshon hora" (the sin of wanton gossip) and "chillul Hashem" (shaming The Name (of God)), but nobody listens, and thus the Day of Atonement finds more sins for the public to atone for.