While very similar in structure, both governments had a myriad of differences. For instance, the Roman government used a modified version of Greek democracy, The Roman voted for two consuls to serve for one year. After they had served their year they were replaced. They were not allowed to be consuls again for ten years. They also could appoint a dictator for six months in the place of consuls in times of emergency.
In contrast, the Carthaginians were governed by an oligarchy, ruled by the few powerful aristocrats. So, unlike Romans Carthaginians had a set of permanent leaders along with appointed generals and admirals but they didn't incorporate the conquered people into their governments' structure, so they did not have the support from people that Rome had from its people.
Militarily, Rome and Carthage were also quite different. The Rome had military groups that were trained to fight on land-based. The citizen could volunteer to join the military and fight as a large joined community. On the other hand, Carthage government had a sea force that includes ships and military personnel. Its government had invested a lot in the navy than the military that was land-based. Carthage, unlike Rome, they could obtain mercenaries through recruitment and also being an empire that was highly commercial, it could hire troops from other counties like Liberia, Spain and even from North Africa.
Romans and Carthaginians also differ in the matter of faith. For Romans, they had Greek gods that were a faith Latin people while Carthaginians believed in Phoenician gods. These suggest that each government allows their religion only to be practiced in the territory.