Indeed, however Carthage is the old power based on trade when the med was made up of small city states. The times had changed and Rome was smack dab in the middle of the medy. On top of that, they had armies and lots of natural wealth.
I find it hard to not have Rome and Carthage but heads when Roman merchants start trying to interfere in Hispanola. Carthage had power because they had money, like Britain and America by the 1900s all Rome had to do was build a descent navy and Carthage goes the way of, well Carthage.
Once Rome was able to look away from Italy, they are gonna start looking to expand there wealth at the behest of someone; and Carthage is THE merchant nation.
Either Rome stays in central Italy battling it out for control of peninsula until Gauls or Germans conquer them or Carthage ends up a Roman vassal.
Unless Carthage decided to say fuck all to the wealth pouring in from the sea and decided to look for wealth inland, then maybe they could get the manpower to fight the Romans. Mercenaries are only good when you have money, so all Rome had to do was build a navy and boom Carthage is without army.
EDIT: Carthage saying fuck all to trade wealth for even a few decades seems ASB, as the city was founded on said wealth.
To be fair, Rome was a city state as well, although by the time of the Punic Wars, they had a far larger citizen base than other Med city-state before, and an even larger military manpower pool from "allies".
Also, the "Carthaginian mercenary army" and "Carthaginian power based on trade wealth", while both partly true, are overblown because our sources tended to be either Greek or Roman, thus anti-Carthaginian by default (this is especially evident in the way Livy described the Carthaginian armies as heterogeneous and dubiously loyal). This is worsened by later modern filters that have not entirely disappeared from public perception, such as the emphasis on Phoenicians as "merchants" which was in harmony with alleged attitudes of another vaguely related Semitic group.