I was hoping someone else would handle this because this is a huge topic in Late Antiquity, but until then I'll give a slight answer.

Carthage was, in late Antiquity, a massive and incredibly important city, whether you are looking at it demographically, economically, intellectually, artistically, what have you. In general North Africa avoided the economic malaise that struck much of the Eastern Mediterranean during the third century (although there was a short decline in public building) and emerged as one of the true heartlands of the empire. The primary products of the province were agricultural and thus cannot be traced, but riding along with the grain shipments was ceramic, the famed red African slip ware that was produced in African workshops and traded along with the grain. And it is everywhere. It is arguably the signature material culture from Late Antiquity and has become the emblem of the period's economy.

Carthage was of course the main node connecting African agriculture to the Mediterranean, and it grew to be perhaps the second largest city in the empire, certainly in the West, after Rome. Granted, many of the most famous monuments, such as the amphitheater, stadium, Zaghouan Aqueduct and military harbor were built during the Early Imperial period but they were maintained during the later period. And we also see signs of private wealth, such as the mosaics on the Villa of the Aviary, which may have been late Antique. This seems to have bled over into a rather vibrant city culture, and St. Augustine said that when he came to his city in his youth "all around me in my ears were the sizzling and frying of unholy loves." So they knew how to party.

And on the topic it became a major intellectual center, and North Africa in general was the center of Christianity in the West. This is definitely something that others can speak of better than me, however.

There is definitely more detail to be given here, but broadly speaking Carthage during Late Antiquity maintained, even enhanced its position as it was largely outside the troubles that engulfed the empire in the third century.

There is a massive amount of quality, up to date research on North Africa, a great, accessible introduction to which is The North African Stones Speak by Paul MacKendrick. Carthage itself is somewhat less well explored, but Serge Lancel's Carthage is still to my knowledge the best survey on the city itself.

Alright, I'll take a stab at this. I'll see if I can flesh out some of the cultural aspects of this question as well as a brief discussion of the Vandal/Roman revival.

As /u/Tiako points out, Africa Province was a particularly wealthy place to be during antiquity. The exceptional wealth of the province caused a rather significant construction boom during the second century. At this point, the cities of the region were expanded greatly. The city center of Carthago Nova had the normal kit of buildings with public life focused (to a certain extent) around the public practice of religion. Like other parts of the the Empire, the Roman authorities synchronized local divinities with their Roman "analogs." Two examples are the Punic deities Milk’Ashtart and Shadrapa who were identified as Hercules and Liber Pater. We know this because there are parallel inscriptions in a temple to the east at Leptis Magna. Thus, we can see a persistence, at least in a small way, of Punic culture despite several centuries of Roman and Latin domination (Rives, Religion in the Roman Empire).

As far as ethnic diversity goes, the aforesaid example suggests a persistence of the old Carthaginian culture to a degree, but I really can't make any guesses to the extent of that aspect of the culture of Africa province. We must not, of course, forget about the various Berber tribes that most likely formed the majority of the population. Augustine was probably a Berber himself. His mother's name was Monica, the only Berber name the make its way into Anglophone naming traditions (Augustine of Hippo, Peter Brown). In Africa province, the Roman state was content to leave traditional structures of power in place and use the cities as centers of agricultural extraction. With this in mind, the Empire coopted local Berber chieftains who would represent the government, and they would, in term, receive citizenship and all of the benefits that attended such a status. We have quite a bit of epigraphic evidence evidence that illustrates this point. Local aristocrats (chieftains turned Romans) erected tombs and monuments with names that demonstrate a mixture of local elite naming traditions mixed with the cognomina of Roman citizens. (Brett and Fentress, The Berbers an excellent book, I might add). These elites would become full participants in Roman urban life, doing the sorts of civic stuff that Roman aristocrats did. That is to say, they saw their local cities (including Carthago Nova) as "mini-Romes" wherein they would compete for offices, form collegia, and invest heavily in monumental architecture (particularly religious architecture in the case of Africa Province) (Hoyer, "Public feasting, elite competition, and the market economy of Roman North Africa"). The most notable example of such a local elite would have to be Septimius Severus. Less accurate histories of the Roman Empire that I have run across assert that Severus was a "black-African" emperor, when in fact he was a Berber from Leptis Magna.

Now, Christianity emerged into this world. The Christian community of Roman North Africa was particularly strong. The region produced some of the finest Latin theologians of late antiquity: St. Cyprian (a bishop of Carthage), St. Augustine, Lactantius (who instructed the children of Constantine in Christianity), and Arnobius of Sicca. By the third century, Bishops had emerged as powerful members administrators of the Christian community. As Cyprian's letter no. 11 demonstrates, the Bishop was able to secure food and support for members of his flock who were incarcerated for not participating in official sacrifices (essentially the crux of most Christian persecutions). Additionally, the Acta Cypriani suggests that at this early date Christians already held land in common(!) and had communal burial grounds as well.

Given the strength of the Christian community in Africa province and the particular ferocity of the Decian and Diocletianic persecution in Africa Province, the African church developed a strong cult of Martyrdom with Augustine remarking in Letter 78 that the region is "...full of the tombs of holy Martyrs."

Given the strength of the Christian Church in Africa province, it is strange how quickly and completely the region converted to Islam. As Ronnie Ellenblum discusses in Frankish Rural Settlement in the Kingdom of Jerusalem, much of the hinterland of the Levant remained Christian at lest until the crusades and I have heard an arguments that it was not until the late thirteenth century with the advent of the Mamluks and Sufi particularly that the region was converted to Islam. Regardless, of the reason, North Africa converted to Islam very quickly. Where we see elements of Christian infrastructure persisting in the Levant, the opposite happens in the Maghreb. Under the Roman revival of Justinian there were about 200 Bishoprics, by 1076 only two remained (Brett and Fentress again).

While there is some debate over the cause rapid de-Christianization I honestly think the traditional assumption to lay the blame at the feet of the Vandals isn't too far from the mark. Yes, as Roman authority declined, independent petty Berber kingdoms emerged, but the Vandals were not able to maintain the economic power of their new home and they attacked the local Christian population, expelling nearly 500 bishops from their sees for adhering to the tenets of the Council of Nicaea (Abulafia, The Great Sea). The Vandals not only disturbed the Nicene status quo, they also demonstrably hurt its importance within the Mediterranean economy. In Iberia, for example, we see a the absence of African Red Slip pottery during this period (Lloret, "Eastern Spain in the Fifth Century in the Light of Archaeology").

Carthago Nova and the rest of the coastal remained important to the Roman Empire even if its economic strength had declined significantly. The Roman attempts to retake the province centered around these cities, and I believe Carthago Nova, in particular, held out for some time.

Edit: I should add that although Diocletian's reforms slowed down Rome's gradual disintegration, it did have long term effects on the unity of the Empire and perhaps the Vandals don't deserve all of the blame, but most of it. I'm happy to discuss Diocletian all day though.