Answer:
Carthage was — or at least, had been for the last couple of centuries — Rome’s only great-power rival in the western Mediterranean. It was a focus for potential anti-Roman coalition building (as Hannibal’s career in Italy showed) and also an economic rival. Carthage’s amazing ability to bounce back from that defeat surprised and dismayed Roman hawks like Cato the Elder (of Karthago delenda est fame).
However, the destruction of Carthage was not a rational piece of statecraft. It was a much more visceral cultural reaction that stemmed from Rome’s self-serving, but nonetheless very fundamental, view of the relationship that had been created by the peace of 201.
On the psychological level, the Roman's believed that the Carthaginians were now subjects, not a foreign power: the Roman view was that the surrender after the second Punic war had made Carthage into a client state whose duty was to take orders. In the Roman view (domestic as well as international) clients were believed to have a duty towards their patrons/conquerors. Failing to follow the wishes of the patrons was not just interstate politics but more like treason. This partly explains the vindictiveness from a Roman perspective, even though it hardly excuses such behavior in our eyes.
There is also the fact that Rome was fighting the Fourth Macedonian War at the same time. From the Roman perspective both of these wars seemed like treachery: defeated enemies who were refusing to remains subordinate were a very scary thing to the Romans, who depended on a complex network of subject people to man their armies and secure their positions. They were far more savage in punishing "faithless" "allies" than defeated enemies: it's not a coincidence that the Romans also leveled Corinth in the same year (146) as Carthage.
The twin destructions of Carthage and Corinth do also reflect the fact that Rome was beginning to realize that it was the dominant power in the Mediterranean world. The Third Macedonian War, a generation earlier, shattered the power of Rome's major rival to the east, just as the defeat of Carthage gave Rome a free hand in the West. That lack of serious opposition seems to have done something to the Roman psyche: It's pretty clear that Polybius, who was a first hand witness to the whole saga, felt that the older generation of Romans were very different than the generation that took Corinth and Carthage. In fact, he he tacked on several more books to the end of his history to follow events after the battle of Pydna, which made Rome 'mistress of the world.’ As those books go on he becomes notably more critical of Roman character and behavior, in contrast to the admiration he showed in his earlier work. He had very complicated feelings about the events of 146, but despite his Romanophilia and his friendship with the Scipios he was painfully aware that something darker and more vindictive had emerged in the Roman psyche after the end of the Hannibalic war.
Step-by-step explanation: