First off, it is necessary to separate Classical (or literary or official or "Elite") Latin from the sermo vulgaris ("common speech" or "vulgar Latin"). Classical Latin was more or less crystalized compared to most languages, and changed little only in minor variations from the Republic down to the end of Antiquity. It was standardized in its orthography throughout the Empire, no matter the location. It was slowly replaced by the emerging Romance languages, at various times and for various tasks, but we can still find late Medieval thinkers writing in perfect "Classical" Latin a thousand years later.

"Vulgar" Latin was the version of Latin spoken in the everyday streets. We know it had minor differences, and that it evolved over time naturally (like any living language). Its vocabulary was slightly different, especially for everyday items (like concepts related to food and cooking, for instance). It existed even during the Republic, but it probably began to diverge from the more static Classical Latin during the High Imperial period.

You are probably expecting me to say that the sermo vulgaris diverged different in different regions, leading to the Romance languages. But it did not, in fact. The source material for vulgar Latin is not great, but from what is available we find an odd homogeneity all over the Empire, from Britain to Carthage to Antioch to the Rhine, all the way down to the end of Antiquity and beyond. The first obvious regional divergences do not start showing up until the 7th or 8th century CE. This common vulgar Latin base is sometimes called Common Romance, the stew from which the Romance languages began to develop in earnest. Why exactly there were not regional variations sooner is not clear to me, since I'm not very knowledgable on Late Antique linguistics. You might try asking an updated version of the question over in /r/linguistics or similar.

Or, I can point you to some basic starting places. A really good book is József Herman, Vulgar Latin (Pennsylvania State Univ. Press, 2000). This is an updated and translated version of his 1967 book (French, le latin vulgaire). He goes through the (lousy) state of evidence, and does a really good job with it. Another is Clackson and Horrocks' History of the Latin Language (Blackwell 2007).