Romans believed that as long as Carthage existed it would always pose a threat because if you defeat them and don't destroy them they will just rise again and want to defeat Rome again so they decided that it was necessary to completely wipe them off the face of Earth, which is what happened in the end, as you can see since there's no Carthage any more.
The war started mainly because of four aspects Militarism Alliances Imperialism and Nationalism. This is because big armies become potential threats to other countries.
