Part of the issue here is that we are not only talking differences in time but in space. The Roman Empire was a large empire which covered a wide area from what we would now call Turkey, Iran and Israel in the Middle East, to Egypt, Spain and England. Before being conquered by the empire, these places would have had their own languages and ways of doing things. Ways of life and languages would have differed quite a bit in each of these places, and despite Roman attempts to spread their culture, each differed quite a bit from life at the center of the empire, Rome. During the Roman Empire, in the provinces, it would be very useful to know at least some Latin because it was the language of the local government and courts, so if you had to have your case heard in the Roman legal system, you would have an advantage. But in their everyday lives, many people in the provinces continued to use whatever their local language was.

We also have accounts of low level Roman officials like Irenaeus complaining about having to learn the local “barbarian language” and know that people who were already bilingual were prized for administration and leading subsections of the army who were often ethnically mixed. For example Herman/Arminius who rose to lead a significant section of the Roman army. (and eventually used his position to turn double agent and drive the Roman Empire from his homeland, but that’s another topic)

In the areas near Rome the local language was Latin, or a very similar Italian language that could be understood by those speaking Latin, so there weren’t the same tensions. The local languages tended to merge and flow with Latin and most people ended up being able to speak at least some of it the same way that many people can easily recognize “proper” BBC English.

Once the Roman Empire fell, however, the situation became more fluid and uncertain. Social structures started changing, and a lot of the benefits of knowing Latin as a second language (at least for the common people) started changing. The leaders during the early middle ages (also sometimes known as the migration period) weren’t that different from those they lead in terms of culture. Most of them spoke a non-Latin local language as their first language, so there wasn’t the need to deal with translation to do basic business with your local government.

However the catholic church was historically part of the government of the Roman Empire. The Roman Empire always had a state religion, at first that was Religio Romana or Roman polytheism, think Zeus, Mars, etc. aBut when one particular emperor converted to Christianity, he set up an official state Christian church. Even after the empire fell, the catholic church kept their Roman customs and focus on Rome for the most part.

For nobles, being able to talk easily with the church was beneficial because they were both powerful entities and could benefit from cooperation. It is also useful for some things, like diplomacy and science, to have a language that is spoken over a wide area, crossing linguistic boundaries. Because Latin was spoken as a second language over a wide area already and because of the church’s influence, Latin became that language. So Latin became the language spoken by the powerful, the church, and scholars.

In Italy itself, things went slightly differently, in many ways modern Italian is still very similar to Latin. A lot of basics have stayed the same, while bits here and there changed in the gradual fashion languages generally do, saying what exact time the transition happened is like trying to ask when exactly Middle English became modern english.

Other languages, the ones we call the Romance languages, also kept some Latin, like Spanish Portuguese and Romanian.

The Catholic Church, however, kept more of the old Latin even in Rome, in part because liturgical or sacred languages, those used mostly for religious ceremonies tend to change less. This is a thing all over the world, Jewish religious services use older forms of Hebrew, Eastern Orthodox churches often use old Church Slavonic, the Coptic Church in Egypt uses Coptic which has some language from ancient Egyptian, Buddhist and Hindu ceremonies use older forms of Chinese and Sanskrit..... you get the idea. Once again goin into the why’s would mean more writing and don’t want to go on too long without being asked. But part of it has to do with needing to interact with older texts.

Basically,

Tl:dr version, this question is complicated because it talks about a huge range of space and time. It’s also complicated because we tend to think of different places when we think of Roman vs. Middle Ages. We then to think of Rome itself when we think of the Roman Empire, and not so much of f the outlying areas, like England, France, and Germany which were mostly on the edges of the empire and backwaters from the Roman POV, and probably never learned that much Latin in the first place, having their own cultures. Where the Middle Ages had a different center and so we tend to think of England France and not so much of Italy. But there also were some genuine differences in how much Latin was used, mostly because during the Roman Empire it was required for legal issues and then afterwards it wasn’t for the most part.