Reconstructing elite Roman dialect, for which we have reams of literature, is difficult enough, but it is virtually impossible to do so for vulgar Latin. That being said, there are a few instances where we can see the dialectical change from classical Latin to Vulgar Latin is retained in modern Italian, one example being the change in the "au" sound to either "o" or "a", which can be seen in the way the Latin Laurentius became the Italian Lorenzo. One of the most amusing pieces of evidence for this is a passage from Suetonius, "Vespasian" 22:

Yet many of his remarks are still remembered which are full of fine wit, and among them the following. When an ex-consul called Mestrius Florus called his attention to the fact that the proper pronunciation was plaustra rather than plostra, he greeted him next day as "Flaurus."

Of course, this should not be taken to mean that Italians and Romans spoke very similarly--there is two thousand years between them--but rather that the change from classical Latin to the modern Romance languages was well underway by the time of the so-called high empire.