I am a retired English teacher who also had some Latin in high school. The evolution of the English language and other languages interests me.
The evolution of Latin is more complicated than I realized many years ago. I am sure I don't understand all the parts to it, but here is my understanding.
The Romans spoke and wrote Latin.
However, as is true of any language, the spoken and the written languages differed. They were not separate dialects and they were mutually intelligible, but the spoken Latin was much more informal, as spoken languages tend to be, and was more flexible and responsive to changes. It gradually lost many of the cases and pronunciations that Classical Latin retained.
The written language tended to be more conservative and changed less.
Eventually the Latin that was closest to the spoken language came to dominate in Rome. Eventually it was so different as to become Old Italian.
As the Empire spread, the Latin spoken by the Roman occupiers merged with the local languages to form subsets of Latin, each of which took on the unique flavors supplied by the local language.
That's why Spanish and Romanian and French and Italian are "Romance" languages, derived from Latin, but also differing in many ways.
I am going to get the dates wrong on this, but just as an example, by the year 500 AD or so, the congregants at a Mass given in some town in France were no longer able to understand the Latin.
English and Latin share a common ancestor in that they are both Indo-European languages, but English is not a Romance language. (English is a Germanic language.) Latin has had a major influence on English, but it came during the Renaissance, not when England was occupied by the Romans.