I think there's a kind of strange convergent evolution with the Catholic Church's use of Latin, but not one that it shares with Ancient Rome.

Rome used Latin aggressively as a means of cultural posturing and spreading knowledge of the Latin language - after all, if the locals understand you, that's less trouble you have to go through to keep them in line. For Rome, they wanted the locals to come to understand Latin to streamline communication, and also to reinforce that their rulers were Roman, not local elites.

However, at the same time, Rome welcomed outside ideas into its institutions, to the point where some of the most learned and respected jurists of Roman law were Syrians, who had a long tradition of (formerly non-Roman) legal education - it was certainly not a question of restricting provincials by insisting on Latin, but by insisting on the way (ie speaking the CIVILIZED, CONQUEROR'S TONGUE) in which they were included. So it was a mixture of practicality (in creating a lingua franca and not needing to use translators all the time) and pride (GO ROME ROME STRONK) to insist on Latin.

The Catholic Church, on the other hand, used Latin in the opposite way, as a means of obscuration. Since Latin, even as early as the 8th century AD, was effectively dead everywhere except Italy, and dead even in Italy by the 10th century AD, there was no question of people 'independently' learning it en masse the way people would learn their own native tongue, or a neighboring language. Thus the Catholic Church maintained a sort of monopoly on the precise meaning of its liturgy and holy texts (especially, in the latter, considering the low rate of literacy).

This was important not just for the usual aspect of controlling the masses, but also because in Christian thought orthodoxy ('correct thought') was core to the eternal fate of one's soul - if everyone could interpret the scriptures for themselves, especially the uneducated (in Church theology), they might get it wrong and damn themselves - and others for all eternity. So for the Church, it was more a question of tradition/legitimacy ("If we admit to linguistic ambiguity, we undermine our claim to being the only source of truth") and maintaining theological unity ("We had a whole murderous schism over one letter in the Greek alphabet; let's not repeat that") by limiting the number of people who were capable, in any real sense, of interpreting theology.

I would compare the Catholic use of Latin to the use of Old Church Slavonic in some Orthodox Churches or Hebrew in Rabbinical Judaism (maintaining a dead language to minimize schisms), or the use of Mongolian in the Mongol Empire (restricting texts and institutional communication from outsiders to privilege an 'inner group' who knows the language); whereas Ancient Rome insisting on Latin has more in common with English in the British Empire (cultural chauvinism + easier recruitment of locals for government and economic work), or even Arabic in the Muslim conquests (language of law and lingua franca, spread to as many as possible).