I know that some people find it problematic to equate the Greek and Roman deities but there’s absolutely no basis for distinguishing between Bacchus and Dionysos. First off there isn’t even a difference in names since Bacchus is just the Latinized form of the Greek Bakchos, which itself is thought to derive from the Lydian Baki, which we find designating both the God and his ecstatic worshipers. This means that the name goes back to the 7th century B.C.E. and is found in all sorts of words for intoxication and ritual madness well before the Classical period.
Secondly, and most significantly, the identification of the two isn’t a case of casual interpretatio graeca whereby an indigenous and originally distinct deity is recognized as possessing similar traits and therefore is claimed to be the same God just with a locally appropriate form. That process may have happened with Liber Pater, Fufluns, and related Italian deities but there was never an indigenous Bacchus. All of the sources from Livy on down—including inscriptions and the archaeological record—make it perfectly clear that he was a foreign import brought from the Hellenic mainland to Rome by way of Magna Graecia and Etruria.
Even once he had gained wide popular acceptance, Bacchus continued to be worshiped according to the ritus Graecus or with Hellenic ceremonial elements intact. In fact this was a big part of what contributed to the Senate’s antipathy for the Bacchanalia—fear of corruption and invasion, that the devotees of the God were setting up their own miniature religious “nation” in the heart of Rome itself. (And of course all that homosexuality and cross-dressing it encouraged.) They outlawed his worship and the keeping of his festivals except in the case of certain hereditary priesthoods and persecuted the Bacchic devotees, which resulted in the deaths of thousands.
This was the first and most widespread form of religious persecution in the ancient world until the Christians came on the scene, and it wasn’t until Julius Caesar repealed the tyrannical legislation that they were free to worship their God openly once more. Of course even during that time Bacchic and Dionysiac cults were plentiful in Italy and Rome, as we can see from things like the records of cult associations, the Villa of the Mysteries at Pompeii, the preponderance of funerary and symposiatic art reflecting his motifs, and the abundance of literary and poetic references to him. They just had to get permission from the Senate and maintain a polite and civilized façade. Caesar’s repeal of the legislation merely ended the appearance and pretense of illicitness surrounding these cults, which nevertheless earns him a special place in my affections.
But really if one has any doubt concerning the identity of Bacchus and Dionysos they need only consult the Latin poets and historians of that period, all of whom were quite certain that they were dealing with the same God—and who are we to argue with them? Or go a little further back and you’ll find Sophokles praising Dionysos as the “Lord of all Italy” and Plato talking about the famous Dionysian festivals celebrated in Magna Graecia. His cult flourished especially in the Apulian countryside, and Southern Italy and Sicily more generally, which is interesting because those regions are exactly where you find Tarantism and related ecstatic cults nearly a thousand years after the fall of the Western Roman Empire, with only a thin and transparently artificial Christian veneer on them. Clearly once he was established there the roots of this God have run deep in the Italian soil and consciousness.