A crushed Rome has some serious inplications for Italy. The Etruscans, far more than Carthage, were in decline at the time. With no Rome, there's a good chance of Celts overrunning larger parts of Italy from the north. In the south, the main rivals for Rome within Italy were for a long time the Samnites. They took control of several cities (including Capua, mentioned by
@dandan_noodles) that Rome later had to wrest from their control. So without Rome, I see the Samnites as the major power in southern Italy, boxed in between a Celtic(-ruled) northern Italy and the poleis of Megale Hellas to the south. If history is any indication, the Samnintes would be most interested in annexing the wealthy poleis to the south, and would be less interested in fighting Carthage or the Celts first (or perhaps at all). In fact, since both Megale Hellas and Carthage claim Sicily, a Carthaginian-Samnite alliance against Megale Hellas may well make sense, resulting in the Samnites controlling all of mainland Southern Italy, and Carthage holding all of Sicily.
After that, if any conflict is going to arise in Italy, it'll more than likely be the Samnites versus the Celts. Carthage remains unbothered for the moment. They'll indeed want to expend their control in the west, which was their OTL goal. Outposts along the Mediterranean coast of Iberia; solidifying their hold on the Balearics, Sardinia and Corsica; annexing far western Hellenic poleis such as Massalia... The popular image of Carthage as some sort of non-militarist "merchant nation" is partially a fiction. Yes, their empire was a trade empire. Yes, they were centred on a strong navy and relied heavily on mercenary forces for action on land. But they didn't lack for ambition. Their OTL empire of city-states didn't come from nowhere. So without Rome to mess everything up for them, the Carthaginians would probably keep expanding, as they had in fact been doing before their OTL rivaly with Rome got in the way of smooth operations.
Of course, Carthage wasn't the highly expansionist-happy power that Rome was, and like
@PhilippeO says, it's unlikely that Carthage will conquer Italy and found a vast empire like Rome did-- but on the other hand, Carthage wasn't some kind of meek peacenik state or something. Without Rome, I see them eventually controlling the Med west of Italy directly, and exerting considerable economic influence over the rest of the Med (since control of Sicily is rather valuable in that regard). The kind of overwhelming economic influence that allows for outright military intervention (though not conquest) when a rival seriously threatens trade profits.
Carthage may not be the power that controls all the Med; it may only be a strong regional player. But it might just be the player that keeps all potential rivals from growing too powerful, thus keeping the whole Med politically divided.