Let's assume the following events happen...
241 BC--The First Punic War ends with defeat for Carthage. The harsh treaty imposed by Rome requires Carthaginian withdrawal from Sicily and several other small island groups in the vicinity of Sicily, but leaves Sardinia and Corsica in Carthaginian hands. Carthage is forced to pay a large indemnity of 1,000 talents of gold immediately, with a further 2,000 talents to be payable in installments over a period of ten years. Another provision requires Carthage to release all it’s Roman prisoners without ransom, but also requires it to pay a ransom to get it’s own prisoners back. The combined effect of these provisions is to effectively bankrupt Carthage.
After meeting the Roman demands, Carthage now finds itself unable to pay the agreed-upon wages of the tens of thousands of mercenaries it employed during the First Punic War. Some 20,000 mercenaries, formerly under the command of Hamilcar Barca (who had resigned his command at the end of the First Punic War), would shortly be returning to Carthage from Lilybaeum, in Sicily. Gesco, the Carthaginian commandant responsible for transporting the mercenaries from Sicily, not wanting to have such a large force of disgruntled mercenaries near Carthage itself, attempted to deploy the mercenaries throughout Carthaginian territory. His plan was to bring the mercenary units back to the capital one at a time, for demobilization and payment. However, the Carthaginian government, believing that the mercenaries could be convinced to settle for less than their agreed wages, delayed in paying and demobilizing the units as they were brought in, which resulted in the eventual gathering of most of the mercenary armies near Carthage.
After the Carthaginian government convinced the mercenaries to withdraw to the nearby city of Sicca Veneria, 170 km south-west of Carthage, taking their families and baggage trains with them, the mercenaries put together a list of demands against the Carthaginians. Hanno the Great, a representative of the Carthaginian aristocracy, meets with officers from the mercenary companies and rejects their demands, claiming that Carthage could not possibly pay such an exorbitant sum due to her post-war indemnities to Rome. Further negotiations quickly break down, and a force of 20,000 mercenaries arms themselves and marches on Carthage. Realizing their error in letting such a large foreign army gather in the first place, and also realizing that they have released the family and belongings of the mercenaries as well and thus had given up a bargaining position, the Carthaginian government has no choice but to capitulate to the mercenary demands.
Not willing to deal with Hanno again, and feeling insulted by Hamilcar for not having met with them in the first round of negotiations, the mercenaries agree to negotiate with Gesco. But given their newly strengthened bargaining position, the mercenaries vastly inflate their original demands, even requiring the extension of the payments to the Libyans whom Carthage had conscripted (who were not mercenaries) as well as other Numidians and to the escaped slaves and the like who had joined their ranks against Carthage. Once again Carthage has no choice but to agree.
240 BC--Two mercenaries, Spendius and Mathos, fearing that once the foreign mercenaries are paid off and leave Africa, Carthage will take it’s revenge on the Libyans and Numidians left behind, organize a rebellion. The Libyan population, discontented with Carthaginian rule, supports the rebels. The city of Utica, which in OTL did not join the rebels, decides to support the rebellion, seeking to finally rid itself of it’s Carthaginian rivals and overlords.
Carthage manages to raise an army of citizens and loyal mercenaries which, under the command of Hanno the Great, moves to attack Utica, which the rebels are using as a base of operations against Carthage. At the Battle of Utica, the rebels inflict a severe defeat on the Carthaginians. Hanno the Great is disgraced, and Hamilcar Barca is placed in command. However, the mercenaries and their Libyan allies quickly move to besiege Carthage itself. Hamilcar is defeated and killed in battle outside the walls of Carthage, and his army is soon trapped within.
239 BC--Carthage falls to the mercenaries. It is razed to the ground, it’s population either massacred or sold into slavery by the victors. Rome, at the invitation of Carthaginian mercenaries which had rebelled on Sardinia and Corsica, immediately occupies those islands.
So Carthage is gone, roughly 100 years earlier than in OTL. There are no Second and Third Punic Wars. This would have massive effects on the future history of Rome.
It could be argued that the Second Punic War, and it's aftereffects, are the root of the disease which finally killed the Roman Republic. The heavy demands on Roman manpower during the Punic Wars...the Second Punic War in particular...and afterward, with the need to maintain control over conquered territory in far-away lands, had forced the plebeian citizens of Rome to leave their farms for long periods to fight. By the time they got back, their farms had fallen into disrepair, and the plebeians started going bankrupt. The landed aristocracy began buying up the bankrupted farms at discount prices, and staffing them with slaves captured during the wars. This caused commodity prices to fall, leading more plebeian farmers into bankruptcy...meaning yet more land was grabbed by the aristocracy, which further fueled the cycle.
Masses of unemployed farmers flooded into Rome, which gave them entry into the legislative assemblies. They tended to vote for populist candidates who offered them economic favors in exchange for their votes. These people became the basis for the Populares movement founded by the Gracchi, whose struggle with the aristocratic Optimates would eventually destroy the Republic.
If there is no Second and Third Punic Wars, this probably doesn't happen. Rome may not expand much beyond Sicily (it's expansion into Africa, Iberia, southern Gaul, and indeed, into Macedonia, Greece, and the Levant, can all be traced to the Second Punic War and it's aftereffects). If Carthage is destroyed by the mercenaries, that expansion may never occur. If so, Rome remains a local power in the western Mediterranean, the plebeian farmers are not needed on long foreign campaigns and so get to keep their farms, and the power struggle between the Optimates and Populares doesn't get started.
With luck, the Roman legions will remain a citizen-based militia force, and there will be no Marius to transform the army into the instrument which was used to destroy the Republic in OTL...a professional army whose loyalty is to it's generals rather than to the government of the Republic. The Republic will continue to gradually reform itself, relatively peacefully, as it had in the period prior to the Punic Wars, giving more power to the plebeians and making the system more representative. Instead of an Empire, you might well end up with the world's first truly representative democracy.
Of course, that's the best case scenario. It could end up much less favorably as well.
Thoughts?