Punic was still spoken in North Africa in the time of St. Augustine; the civilization thrived into Late Antiquity as is, and disappeared only in the chaos of successive Vandal, Byzantine, and Muslim conquests.
Carthage had a real shot at replacing Rome as the dominant power over the whole Mediterranean. The two shared many key strengths, such as a position in the center of the Mediterranean basin (decreasing transport costs), a republican state that could maintain legitimacy in the face of repeated defeats, immense manpower resources, highly flexible infantry, and a naval strategy focused on decisive battle. Aside from Rome, the other contenders, Egypt and Syria, were monarchies where the ruler's legitimacy depended on his success; they had limited Greek manpower, their tactical formations were highly vulnerable in broken ground, and their fleets were specialized for naval siege warfare.
Defeating Rome doesn't make Carthaginian hegemony inevitable, but it does give them an opportunity almost as good as the Romans had. Like Latin, Punic would probably not replace Greek in the East, but the use of Punic as a lingua franca in Carthaginian armies and the proliferation of Punic colonies would suggest a dominant Carthage would see its language spoken quite widely outside its region in North Africa, and over enough time would probably take on a life of its own.
