The Roman Republic's army was based in its citizen soldiers, as opposed to Carthage, whose army saw far greater reliance on mercenaries. Carthage, with its more far-flung empire (with holdings across the larger Mediterranean world), was far more reliant on its naval power, while Rome was still, as of the First Punic War, a land-based empire. However, one of the critical advantages of the Roman military (one previously foreshadowed during the earlier war against the Greek commander, Pyrrhus of Epirus) was its manpower reserves and its ability to recover from military losses that (for many other states of that time) would have been catastrophic.
This dynamic of Roman land superiority against Carthaginian naval strength was a critical theme in the First Punic War, and it actually inspired Rome to invest in the creation of its own navy. One of the critical tactics of this first stage in the war, from the Roman perspective, was its reliance on boarding enemy ships rather than outmaneuvering them (a strategy that was to Rome's advantage).
The Second Punic War, on the other hand, was largely defined by Hannibal's invasion of Italy, with the intention of defeating Roman armies in direct military confrontations and turning its allies in Italy against it. Ultimately, here we see Rome's manpower reserves utilized to their full effect as Rome endured the defeats (most notably the defeat at Cannae, one of the most catastrophic defeats in all of Ancient History) while refusing to surrender.
Hannibal, meanwhile, lacked the resources to siege Rome itself. Thus, despite his victories, the war turned increasingly against him, as Rome began utilizing a Fabian strategy, refusing the direct battles that Hannibal sought, and instead preserving its forces, shadowing Hannibal, to maintain the loyalty of its allies. Thus, Hannibal was forced into an increasingly precarious situation, with his initial strategy now working against him.
The final turning point of the Second Punic War came with the emergence of Scipio, who would lead Roman armies to victories in Spain and later Africa. In the process, Hannibal was forced to abandon the Italian campaign to address the invasion of Africa, where he would be defeated in the Battle of Zama. The Second Punic War would end in a decisive victory for Rome, and these conflicts with Carthage would signal its emergence as a major power in the Mediterranean world.