After the First Punic War, Sicily became Rome's first "overseas province." Rome was also able to gain control of both Corsica and Sardinia, two islands located of the coast of Italy. This is significant because it demonstrates how powerful Rome's forces were and that with little experience they were able to develop a well trained military. It also is important because it expanded the Roman Empire, which benefitted trade routes and made it easier to access trading since it was surrounded by water, so it was easy to travel to and from other cities and countries.
The Punic Wars gave Rome experience in combat, especially at sea (in the navy) which they never really had prior to the wars. It also taught them the skills of training. Finally, the wars provided wealth to Rome with all the money Carthage had to pay in war debt. Learning all of these tactics and items were significant because they led to the continuation of building the Roman Empire. Specifically, with the experience and wealth Rome had gained they were later able to be victorious in the Macedonian Wars.
Lastly, the Punic Wars impacted Rome because by defeating Carthage, Rome became the dominant power in the Mediterranean Region, since before the wars Carthage was the most dominant. This is significant to Rome, because they received more control and had the most power in the Mediterranean Region.
Carthage
Carthage was completely destroyed and was just ashes for more than 100 years, which is significant because after 700 years the city had been teared to pieces. Also all of the remaining Carthaginian citizens who survived were put into slavery. So by the end of the Punic Wars Carthaginians were either dead or slaves.