Well, if the Celtic Senones under their king-chieftain, Brennus, went the whole hog and just massacred all the Patrician and Senatorial families, as well as massacred or enslaved the remaining Plebeian class within the city, instead of just forcing tribute out of them, they would have resumed with their initial aim of conquering the Etruscan city of Clusium, whom they were at war with before the Romans intervened.
The Senones were really after land in Etruria (Tuscany), and had they definitively destroyed the Roman Republic, would have sought to establish their tribe's hegemony over the cities of Etruria and Umbria in the east, and assuming they maintained some kind of political unity among themselves, would have established a kingdom with Clusium as their capital. Otherwise, Etruria, Umbria and maybe parts of Latium, would have descended into a hodge-podge of separate, warring, Celtic -dominated city-states.
Either way, there is no guarantee that some other city-state federation like the one Rome dominated in OTL that would arise to establish a Mediterranean-centred empire.
More likely the next most dominant force for civilization in the western Mediterranean in the absence of Rome will either stem from the Greek city of Syracuse in Sicily, the Punic commonwealth of Carthage, or one of the tribal confederacies in Gaul.