Yes and no.

For the most part, students studying Latin in most of North America and Europe are studying "classical Latin", this is Latin based on what was written in Latin. These authors include Ovid, Caesar and Cicero, among many others. Of course, Latin did change over time like every language, and what we now consider classical Latin is really just the language that the higher classes used when writing. The language that was spoken by everyday citizens is known as "vulgar Latin" (vulgar meaning something like "everyday", and should be understood in a "non-offensive way"). Vulgar Latin is what eventually turned into the modern Romance languages--French, Italian, Romanian, &c (they are called Romance languages because they are descended from the language of the Romans, it's really just coincidence that today we happen to think these languages are romantic in terms of love, although the word romance does come from Roman I'm not sure exactly what the lineage is).

Classical Latin managed to stay pretty consistent throughout history. This is true of written languages in general. Formal English writing from hundreds of years ago is pretty much indistinguishable from modern formal English, even though the accent and dialect of the people around the author are very much different.

In the fourth century, a Latin translation of the Bible was made, but it was translated into vulgar Latin (this translation is called the Vulgate) instead of classical Latin. This was intended to make it readable by as many "commoners" as possible, not restricting access to a special papal class. At this point, vulgar Latin was still fairly similar to classical latin in writing (the modern romance languages hadn't really started forming yet). Ironically, about a thousand years later the Catholic church started using this as their official Bible, and it worked well for them because vulgar Latin was no longer spoken among the working class. This let the church keep the Bible as a special resource that "commoners" did not have direct access to, but access to the Bible isn't the topic of this discussion...

Classical Latin wasn't really a thing at this time. A lot of classical Latin is based on modern reconstruction of how Latin was probably spoken in the classical age. As time goes on, words become pronounced differently so it can be hard to figure out how a word was pronounced at a certain point in history.

So, the Catholic church was working with a Latin tradition based on vulgar Latin spoken around Italy. Today we call the dialect of Latin being used by the Catholic church Ecclesiastical, which is pretty similar to classical Latin except in pronunciation, where it tends to sound distinctly Italian.