For the Carthaginians, their empire can best be styled as a thalassocracy, which means rule of the sea (think Athens and Venice). They didn't make many land claims, preferring to have some outposts, while maintaining a dominant position in the sea in order to dominate trade in the region. They might have sought to expand down the western coast of Africa (they had claims to the west of the Strait of Gibraltar) had they not run head first into Rome.
The Punic Wars, altogether, took over 100 years to conclude (264-146 BCE), but it was the first one that really set Carthage back the most in regards to its goals of controlling the sea. While the Second Punic War (218-201 BCE) is usually more popular to talk about (Hannibal and all), the First Punic War (264-241 BCE) is interesting because Carthage was probably stronger than Rome when the war broke out (Carthage was larger as well). The war was over control of Sicily, which Carthage had and used to project its power in the region. Rome, being the only other powerful polity in the area, didn't like this, especially because Sicily was so close to Italy. When the war started, Carthage had a significant naval advantage, but as time wore on, Rome was able to equalize the playing field by building up its navy and developing new tactics. The war ended with Rome defeating Carthage, annexing Sicily (other than Syracuse), and basically ending any potential for Carthaginian hegemony in the region for the near future. If Carthage had won, we might have seen a history where Carthage was the dominant power in the Mediterranean, and not Rome (though we can never really know).
The Second Punic War ended Carthage as a regional power and the Third one ended Carthage altogether. At this point, Rome was the only power left in the region, which allowed it to expand throughout the Mediterranean. We see this occur up to Trajan, when Rome conquers Mesopotamia.
If Rome were so powerful, why wouldn't they try to expand down the African coast? Mainly because it was probably expensive, the people living there were not an immediate threat, and Rome was already too large for its own good. For the last point, Hadrian, Trajan's successor, gave up Mesopotamia when he became emperor because it was too difficult to defend. Marcus Aurelius defeated various Germanic tribes in the Marcomannic Wars and could have expanded into modern Hungary and Czech Republic, but these plans were given up by Commodus because the Danube was a good natural defense (compare this to Dacia, which is modern Romania, which was above the Danube and notoriously difficult to defend, eventually being given up by Aurelian I think). Rome had more pressing matters (Parthians, Germans, internal crisis, etc) to attend to, rather than expand down the African coast.
As a final point, the East was much more appealing than the South for would be conquerers. Parthia was Rome's most consistent and powerful enemy, and Rome probably knew of the gains to be had if they had conquered Parthia. Finally, conquering Parthia would have been a contemporary equivalent of Alexander conquering the Achaemenids and many Roman leaders sought to be like Alexander.
In all, Carthage maybe could have if Rome didn't literally destroy them in three wars and Rome had more pressing matters including dealing with Parthia and the border regions.