Roman graffiti from Herculaneum and Pompeii in particular shed some light on early vulgar Latin, suggesting that already that the regional homogeneity of the vulgar register was already breaking down by the time of the Plinian eruption. Given the large volume of uncovered graffiti, it is fairly easy to discern several trends, notably the loss of written dipthongs (æ->e, oe->e; comparably au->o), losses of final unaccented consonants and medial vowels.

http://ancientgraffiti.org/about/ is an excellent resource specifically for Herculaneum and Pompeii, but it also links to broader collections to which the project has contributed.

There are interesting aspects of graffiti throughout the Roman Empire. Children (or exceptionally short adults) practised writing on walls; some taller people's graffiti showed not just literacy but familiarity with Virgil and even decent command of Greek and other second languages. Conversely, numerous graffiti are supporting evidence for partial Latin literacy among speakers of other languages, even among celtic-language informal epigraphers in the west and northwest in the first decades CE. It seems likely that these influences "accented" day-to-day Latin, perhaps comparably to https://en.wikipedia.org/wiki/Singlish .

A couple of centuries later and one could see a clear split between the classical register and early Vulgar Latin. Appendix Probi is interesting, and some examples are listed at https://en.wikipedia.org/wiki/Vulgar_Latin#Evidence_of_chang... which also draws some comparisons with Romance languages. (cf. https://en.wikipedia.org/wiki/Appendix_Probi )

Latin and the Romance languages are indeed very different, but their similarities are much stronger than just vocabulary. The Romance languages have lost noun cases and gained prepositions, fixed word order, and definite articles, but they retain noun genders (though without neuter), remnants of the case system in pronouns ("je", "me", "moi"), many of the same verb tenses, and the subjunctive/indicative moods, to give a few examples.

Anyway, my point is that it's an error to say that Latin "stopped changing". Other languages have changed similar amounts: modern Americans can't understand Beowulf, modern Greeks can't understand Homer, and modern Chinese people can't understand Confucious, but nobody would claim that English or Greek or Chinese died and stopped changing. The fact that people call Latin, but not English, a "dead language" is purely due to the fact that the different stages of English all happen to be called "English".

In an alternate world where Latin was exactly the same as it was in the past, and Italian is exactly the same as it is now, but Latin had never spread outside of Italy, I suspect that we would today call Latin "Old Italian" (or perhaps we would call Italian "Modern Latin"), and nobody would be having this discussion, despite the scientific/linguistic facts being identical to what they are in our reality.

> The fact that people call Latin, but not English, a "dead language" is purely due to the fact that the different stages of English all happen to be called "English".

I don't think this is the case: the language that Beowulf is written in is normally referred to as Old English. Chaucer is Middle English. Shakespeare is Early Modern English. William Makepeace Thackeray is Victorian English.

IMO, it's reasonable to assert that each of these are "dead" in some meaningful way: even Early Modern and Victorian English, despite their intelligibility, are simply not spoken by any group of current-day English speakers.

The difference between Beowulf English vs modern English and say Italian vs Latin is not the same at all

You're right, Italian is much closer to Latin than English is to "Old English".

Do you care to elaborate?