^ ^ ^
Absolutely right. Carthage's system of government could never have competed against Rome. Only a change in government could have saved Carthage from eventual collapse, and there were only really two men - if we include Hannibal's brothers, who were less talented than he was, but certainly far more able than the average Carthaginian, we get five(?) - capable of doing that, neither of whom ever tried. The only real chance of Carthage defeating Rome without changing its system of government was Hannibal's army.
Assuming that Hannibal actually conquered Rome, then what? Hannibal is now clearly the most powerful man in the Mediterranean world. He has a loyal army, the most experienced and skilled in the entire world at this point, and the prestige that comes from avenging the losses of the First Punic War. The Carthaginian people are unlikely to praise him, heap glory upon him and be thankful for his skill, as the Romans were to their great generals. They would fear him. They would hate him. And they would try to eliminate him.
They would likely attempt an assassination. If this succeeded, Hannibal's brothers would take their mighty army and make their home city pay for its actions. If it failed, Hannibal would do the same. I doubt there would be any sacking, rape or looting of Carthage, as both Hannibal and his brothers were too disciplined to let their troops do this to their home city, regardless of their feelings for its leadership. But essentially, you now have the Barcas as an oligarchic military leadership over Carthage.
I would assume that this opportunity would be seized by every group in the Carthaginian Empire, particularly Rome, which would rebel. But I doubt the Barcas would allow such rebellions to get far, and would probably leave a sizeable force at Capua's disposal to end any revolts in Italy anyway, let alone their sizeable forces in Spain. There is the possibility that Hannibal may move his capital to Cartago Novo, but I think he's too strategically minded for this. If anything, he may take Syracuse and turn it into his capital.
Carthage would probably consolidate its position, and continue its expansion, but Hannibal was no politician, and most of his conquered territories would chafe under his rule. I don't see a Carthaginian Empire such as this lasting long beyond Hannibal's death. There is the possibilty that an able man in the Barca family, or even an usurper, might institute reforms which prolonged the life of the Empire, but I honestly don't see Carthage surviving as a whole for long... Unless it adopts a system very similar to that later used by Augustus, which took a lot of change for the Romans to bring about, let alone a vastly different people like those of Carthage.
All in all, unless Hannibal actually genocided Rome, I still see it as eventually becoming the strongman in the region, although this would likely take a lot longer, be less effective and total, and probably not last as long either. But the barbarian invasions would still have come, and I don't see the world being terribly different from what it is today.