Was it to take over trade routes or get more people or just plain hatred? Or have we just embellished the legend?
Fear. The Carthaginians were Rome's greatest fear. This was despite the fact Carthage had become a shell of what it had been, I mean it used to control the Western Mediterranean.
Even though the Romans defeated the Carthaginians twice and took large and valuable sources of income it always seemed to recover. Cato The Elder, a man who hated the Carthaginians, ended all his speeches in "Furthermore, I believe that Carthage must be destroyed".
This is after the Carthaginians asked the Romans for the right to form an army against the encroaching Numidians in which Cato was sent to speak with the Carthaginians and saw their wealth, he refused to allow them to rally their army, Carthage declined to accept the Roman order and mobilized its army regardless. Which started the Third Punic War which ended with Carthage being burned to the ground.
I hope that clears it up, hopefully someone else might be willing to expand on that point.