Traditionally, Judaism was not an evangelistic religion (and so it remains among 'mainstream Jews') - Jews did not believe it was their mission to spread their faith to the world, nor is there any mention of such a mission in the Hebrew Bible according to the traditional Hebrew rendering. "And you shall be a Holy People for Me" Shemot 22:30 (one of many)
To paraphrase the language of the Talmud:
Do we not have enough problems with the Jews we already have?
Moreover, the Talmud and Maimonides specifically enumerate certain commandments specifically for non-Jews to follow and non-Jews are promised heavenly reward for obeying those commandments. They are not required to embrace Judaism to achieve "salvation". See: According to Judaism, as expressed in the Talmud, the Noachide Laws apply to all humanity through humankind's descent from one paternal ancestor, the head of the only family to survive The Flood, who in Hebrew tradition is called Noah. In Judaism, בני נח B'nei Noah (Hebrew, "Descendants of Noah", "Children of Noah") refers to all of humankind. The Talmud also states: "Righteous people of all nations have a share in the world to come"
As for converting to Judaism:
Our Rabbis taught: If at the present time a man desires to become a
proselyte, he is to be addressed as follows: ‘What reason have you
for desiring to become a proselyte; do you not know that Israel at the
present time are persecuted and oppressed, despised, harassed and
overcome by afflictions’? If he replies, ‘I know and yet am
unworthy’, he is accepted forthwith, and is given instruction in some
of the minor and some of the major commandments.
Maimonides: Mishneh Torah: Hilkhot Issurei Biah 14:2:
We inform him of the fundamentals of the faith, i.e., the unity of God and the prohibition against the worship of false deities. We
elaborate on this matter. We inform him about some of the easy
mitzvot and some of the more severe ones. We do not elaborate on this
In addition, circumcision is required - Talmud, Maimonides, Supra - obviously no simple matter for an adult male, particularly before modern anesthetic techniques had been developed. Thus it provided a good reason not to convert to Judaism.
Christianity, on the other hand, was evangelistic from its very early days: The Gospel of John then points out that Jesus' disciples were baptizing more people than John (John 4:2).
So it's no surprise that Christianity "won" when it came to converts.
As far as the Roman pantheon is concerned, I have little knowledge from primary sources. But from both Talmudic and historical sources that I am familiar with, it appears to me that there was no formally organized religion or movement of "The Roman Pantheon", whose mission it was "to spread the faith", etc, but simply a collection/hierarchy of various deities and their accompanying customs and forms of worship that became part of the cultural fabric of Rome. So, as classical Roman culture faded, so did the Roman Pantheon.
In short, there really was no competition to gain converts. Christianity won by default.