You can stop here if you just want the basics. It gets more interesting.
I've often puzzled over whether I should be using the term Carthaginian (a civic label referring to the city of north Africa the Romans called Carthago) or Punic when referring to the people of northern Africa fighting in the wars with Rome known as the Punic Wars, since Punic can refer to cities elsewhere, like Utica. Finally I found two articles that elaborate my confusion and may help you, too:
"Poenus Plane Est - But Who Were the 'Punickes'?"
Jonathan R. W. Prag
Papers of the British School at Rome, Vol. 74, (2006), pp. 1-37"The Use of Poenus and Carthaginiensis in Early Latin Literature,"
George Fredric Franko
Classical Philology, Vol. 89, No. 2 (Apr., 1994), pp. 153-158
The Greek term for Punic is Φοινίκες 'Phoenikes' (Phoenix); whence, Poenus. The Greeks did not distinguish between western and eastern Phoenicians, but the Romans did -- once those western Phoenicians in Carthage began to compete with the Romans.
Phoenicians in the period from 1200 until the conquest by Alexander in 333, lived along the Levantine coastline. The Greek term for all the Semitic Levantine peoples was Φοινίκες 'Phoenikes'. After the Phoenician diaspora, Phoenician was used to refer to Phoenician people living west of Greece. Phoenician was not, in general, used of the western area until the Carthaginians came to power (mid-6th century). [See Founding of Carthage.] The term Phoenicio-Punic is sometimes used for the areas of Spain, Malta, Sicily, Sardinia, and Italy, where there was a Phoenician presence. Carthaginian is used for Phoenicians who lived in Carthage. The Latin designation without value-added content is Carthaginiensis or Afer, since Carthage was in northern Africa. Carthage and African are the geographic or civic designations.
Prag writes:
"The basis of the terminological problem is that, if Punic replaces Phoenician as the general term for the western Mediterranean subsequent to the mid-sixth century, then that which is 'Carthaginian' is 'Punic,' but that which is 'Punic' is not necessarily 'Carthaginian' (and ultimately all is still 'Phoenician')."
In the ancient world, the Phoenicians were notorious for their trickiness, as is shown in the expression from Livy 21.4.9 about Hannibal: perfidia plus quam punica ('treachery more than Punic').
