I know that in 146 BC the Third Punic War ended with the Romans sacking Carthage, according to legend destroying the city and sewing the earth with salt so nothing could ever grow there again. However I also have heard that it returned as a city by the time of the Imperial era. How did this happen? Was the story about Rome razing the city false? Did some new people come in, rebuild on the same site, and use the same city name? Was a different city nearby renamed Carthage?
As you mentioned, Carthage was destroyed by Romans in the aftermath of the Third Punic War, and a commission was sent by the Senate to ensure that Carthage would remain uninhabited. The land was also cursed.
There were a few attempts to resettle the land. Gaius Sempronius Gracchus wanted to develop a colony called Junonia on the site of Carthage in 123 BC but this eventually was unsuccessful. Pompey in 81 BC renewed the curse on the site of Carthage and a later attempt to sell off Carthaginian territory in 64 BC was rebuffed due to it being sacrilegious and that a new Carthage could pose a threat to Rome. In 44 BC Julius Caesar would, according to Appian, have a dream that resulted in him ordering the colonisation of Carthage, and in 29 BC Augustus began to reconstruct the city, and it became the capital of the province Africa Proconsularis. It became one of the most important cities of the Roman Empire before becoming the capital of the Vandal Kingdom. Belisarius reconquered it. Hasan ibn al-Nu'man, the Islamic governer of the Maghreb, would eventually destroy it 698, after previously capturing Carthage but losing it to a surprise attack by a Byzantine fleet. The regional capital would then shift to Tunis, as Carthage became a village.
More recently, as Tunis grew, it incorporated Carthage as a suburb. It is now a rather upmarket and pricey part of the city.
Source: Carthage Must Be Destroyed by Richard Miles, 2010
You may find some of this thread I saw yesterday interesting, even though it doesn't go into what the Carthaginians did afterwards:
One thing I found interesting is that Scipio apparently did not actually salt the earth!