Why did the original Carthage never become a Roman city?

First of all, Carthage did not fall in the First or Second Punic Wars. The Carthaginians were defeated twice, and compelled to surrender to particularly harsh terms the second time, but the City of Carthage itself was not conquered. Keep in mind that Carthage was not some run of the mill city-state, but rather the capital of a far flung maritime empire.

First Punic War

It took years of intense fighting for the Romans to merely capture Sicily. After 23 years of warfare Rome was exhausted; it had lost numerous men, and the ship building push exhausted Roman finances. A conquest of the Carthaginian heartlands in Africa was simply not a realistic possibility under such circumstances. Even if Rome could marshal the resources for such an invasion, the public would have been in no mood to stomach the idea.

Rome was thoroughly exhausted, and the moderate terms of the peace prove that Lutatius and the senate were well aware of this unpleasant fact. It had been difficult enough to build and man a strong new fleet, though in spite of Italian war- weariness Lutatius and the senate had succeeded in wringing this last effort out of an exhausted Italy; but, if they had only hinted at the possibility of invading Africa, they might have set off a revolt.

_{Thiel, Johannes Hendrik. A History of Roman Sea-Power before the Second Punic War. North-Holland Publishing Company, 1954.}

Second Punic War

Carthage was most certainly unlikely to survive determined Roman siege after the decisive Battle of Zama. Conversely, however, a complete Roman conquest could not have been effected without this siege, which would have been a lengthy and costly affair. Remember that decades later, Carthage held out for three years against a much stronger Rome in a third war.

At this point however, the Roman state was exhausted. Unlike the first war, this time Roman financial resources were depleted barely a quarter into the fighting.

By the end of 214 Roman resources were exhausted and no money was available to pay necessary expenses ... the third instalment [of repayments for the special levy of 210], due in 200, could not be paid and ager publicus was given in lieu. Little progress seems to have been made also with the repayment of orindary tributa; one tributum was repaid in 196, but 25 1/2 tributa remained to be repaid with the booty brought bak by Cn. Manlius Vulso in 187.

_{- Crawford, Michael H. Roman Republican Coinage. Cambridge University Press, 2001.}

Furthermore, Roman victory had been achieved at a high demographic cost. The late Peter Brunt for instance estimated that Roman citizenry losses amounted to 50,000 from 218 to 215, and 75,000 from 214 to 203. This represents a severe demographic shock - the Roman census of 234 registered just 270,000 citizens. The 204 census near the conclusion of the war had fallen to 214,000. Factoring in natural manpower increases, one scholar has even calculated the total losses to be 170,000.

[N]o fewer than 170,000 adult male citizens may have been killed or otherwise lost between 218 BC and 203 BC ... The 'net' loss of manpower resulting from the Second Punic War may therefore have been of the order of 120,000 men.

_{- De Ligt, Luuk. Peasants, Citizens and Soldiers: Studies in the Demographic History of Roman Italy 225 BC-AD 100. Cambridge University Press, 2012.}

Thus, after years of bloody fighting, it is easy to see why the Romans accepted the Carthaginian surrender instead. The severe terms of peace provided Rome with a huge, sorely needed annual indemnity, and practically eliminated Carthage as a military threat. There was no need to physically sack the city, when it was willing to disarm and furnish Rome with its wealth.

Soon after the war ended, Carthage even helped Rome's military adventures:

Just one year after the end of the war, therefore, the Carthaginians were able to supply 400,000 bushels of corn to Rome and to the Roman army in Macedonia. This was followed in 191 by the offer of a gift to Rome of 500,000 bushels of wheat and 500,000 bushels of barley for its war with Antiochus. Twenty years after that a further 1 million bushels of corn and 500,000 bushels of barley were sent for Roman forces fighting in Macedonia.

_{- Miles, Richard. Carthage must be destroyed: The rise and fall of an ancient civilization. Penguin, 2011.}

Third Punic War

Contrary the question's premise, it was in fact nothing out of the ordinary for Romans to raze a city and enslave its inhabitants. In the year 146 alone, the same cruelty was inflicted upon another venerable city, Corinth. Like Carthage, the Greek city was sacked and its inhabitants either slaughtered in the ensuing massacre or sold into slavery.

In fact, Rome had great incentive to sack the wealthy city. The city's economic resurgence was a chief cause in Roman unease in the first place, and its success made it also an enticing opportunity for immense profits from plunder.

The sacking of two of the richest port cities in the ancient Mediterranean was, for one thing, a hugely profitable business. Both cities were brutally stripped of their wealth, and their works of art were shipped back to Rome ... slave auctions and the seizure of a large swath of previous Carthaginian territory, which now became public land owned by the Roman state, unequivocally contributed to a massive infusion of wealth into both public and private Roman coffers.

_{- Miles, Richard. Carthage must be destroyed: The rise and fall of an ancient civilization. Penguin, 2011.}

Ultimately, Rome had no need for yet another client. As Rome's greatest enemy, the complete and utter destruction of Carthage was not only profitable, but also a symbolic gesture on the part of Rome.

In a sense, the Roman did absorb the Carthaginian people. Apart from the numerous enslaved Carthaginians now part of Roman society (even if at the lowest rung), virtually all Punic settlements were brought under Roman rule. The Punic community was not eradicated in North Africa, and indeed the descendants of Carthage eventually prospered as traders in the Roman system.

One of them, Septimius Severus, even became imperator in the second century.