After this, there could have been very little perceived threat left; not only were the Carthaginian's surrendering rather peacefully, but they were even giving up their means of waging war effectively. The giving up of weapons in an age when manufacture and shipping -- the two methods by which any commodity, military or otherwise, can be obtained -- took an extended period of time meant that the Carthaginians were showing themselves to desire peace not only in the short-term, but as a general social principle.
Their submission to the Romans, then, should have been the end of the war. If the reason behind Rome's military invasion of the Carthaginian territory was the possible threat the area presented to Rome, then its disarmament would have solved that problem. The Romans refused to let the issue go, however, demanding that the entire city of Carthage be destroyed right to the ground.
It was the Carthaginian refusal to do this that led to the prolonged siege that formed the bulk of the Third Punic War. It is also this demand of Rome's that makes the true motive behind this war clear; it is politically much more complex than simply having a threatening neighbor in close proximity, but also comes down to simple greed.
It of course seems unfair that the Romans should demand the utter destruction of what at the time was an incredibly large, populous, and prosperous city. It might also seem like the Romans could have little practical reason for desiring this. Even a basic understanding of the mercantile realities of the time, however, can make it very clear why Rome wished to remove Carthage from the map. Ports are incredibly important even today, handling a huge quantity of imports and exports in countries around the world, and facilitating trade by serving as points of entry (or exit) for the vast amount of goods not directly available in any given area. Today, air shipping and railway systems are also able to move large quantities of goods, but in Roman times boats were the only efficient way to move a large amount of cargo from one area to another . This made waterways and the cities on them important commercial and financial centers due to the control they could exert on trade, and Carthage was one of the biggest ports on the North African side of the Mediterranean -- and Rome didn't like not being a part of it.
One of the offers Rome made to Carthage was to have the city moved ten miles inland, which would have been just as bad as a destruction of the city as its usefulness as a port and therefore trading center would be ruined.
It is also fairly clear that Rome did not expect Carthage to comply with this request, as they had never made any similar demands.
Either way, the real reason the Romans went to war with Carthage was to remove it from competition.
At this point in the progress of Western civilization, the Roman Empire was still on the rise. The Republic was still strong, but the Triumvirate and eventual dictatorship that typified the latter half of Roman rule were only a century away. This was a period when Rome was consolidating not only the vast territories and peoples of the known world, but also the power that had over them politically, militarily, and perhaps most importantly economically. In order to increase this dominance, it was willing to employ virtually any tactic available. Such tactics included, in the case of the Third Punic War, overtly stating their ultimate aim in the form of a demand that, at least to some, justified the continuation of the war for an additional three years of siege that culminated in razing a vast city to the ground after a bitter house-by-house battle. That is, the Romans very possibly demanded that the Carthaginians destroy and/or move their city knowing that they would not comply with this component of a peace agreement, and could then justify their own destruction of Carthage.
The complex interplay between Rome, Carthage, and the Numidians under King Massinissa also cannot be disregarded when considering the lead up to the Third Punic War. The situation between the three powers interested in the North African region does not really affect the motives of Rome's war, but it does strengthen it. Rome already had basic imperial dominion over the area, as evidenced by their ability to create the terms of peace and arbitrate disputes. But the power structure here was not solidified, and various factions existed in Carthage and its environs that supported the various competitors for dominance.
Reading into the complex political situation created by these factions and Rome's desire to continue increasing its consolidation of power allows one to pick out even stronger reasons for Rome to go to war.
The newest of these powers, and in many ways the one with the fastest gains in the area, was Massinissa. His continued appropriation of Carthaginian lands and agricultural techniques during the period between the Second and Third Punic Wars provided him with large surpluses in grain and other commodities, which were made into gifts to the roman Empire -- specifically going to feed the massive and widely dispersed Roman army -- making him a definite favorite of the Romans.
Massinissa seemed somewhat less effective at creating agriculture in his own undisputed territories, however; he was quite dependent on the lands rightly held by the Carthaginians if he was to keep supplying Rome in the fashion he was.
This is likely the reason that Rome's ambassadors, attempting to work in secret but unable to keep their obvious preference for Massinissa from being observed, consistently ranted Massinissa the Carthaginian lands his people (and/or military) occupied. As long as Massinissa retained control of the fertile land in the area, Rome could be assured of a reliable food supply for their army. This makes the idea that Carthage was somehow a threat to Rome somewhat easier to swallow, as the threat of Carthage taking back agricultural lands that belonged to them would make Rome less able to wage ware. But even this threat was too insignificant to be given any real credence. Greed and the desire to get rid of competition were still the driving forces behind Rome's decision to destroy Carthage, its just that this greed was not simply directed at Carthage's prominence as a port city. It also included their claim to the surrounding agricultural lands, and the fact that they were becoming a center of anti-Roman leanings.
These leanings were not played out militarily for lack of the true means to wage war effectively against Rome. Though the Carthaginians managed to prolong the siege for three years, the entire city had to be employed in its defense in order to make this happen, and it still eventually fell. Civil unrest, however, is inefficient, and having more than one powerful governmental influence in the area is likewise detrimental to the speedy and effective workings of an Empire. By not only destroying Carthage, but in enlisting Massinissa's aid in the endeavor, Rome consolidated power in the region not only by removing a competitor, but also by greatly strengthening the position of an ally who already had large popular support and a track record of freely providing Rome with its excess prosperity. This was exactly the sort fo government Rome needed in the are in order to most effectively use its resources, and this is the same basic model of government they employed throughout the Empire. Rather than micromanaging every detail of life in other areas, they simply wanted the available wealth and resources with as little trouble as possible. Making the Numidians under Massinissa the last remaining power (aside from the Romans themselves) accomplished exactly that.
Creating the appearance of a threat to justify war, replacing an unfriendly government with one designed to help a foreign nation's interests -- the causes and effects of war have changed very little over the past two thousand years. No matter what the specific circumstances, war always comes down to wealth and power. Rome did not really feel threatened by Carthage, it simply wanted the resources available in that geographic location. This historic pattern has been repeated countless times since.
Appian's Roman History, Book VIII. pp. 515.
Imperialism and Self-Defense. "Chapter 4: The Wars of 219-70 B.C." pp. 236.
P.G. Walsh. "Massinissa." The Journal of Roman Studies, 55(1/2), pp. 149-60