Basically, Punic refers to the Punic people, i.e., the Phoenicians. It is an ethnic label. The English term 'Punic' comes from the Latin Poenus.
Should we 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? Here are two articles that elaborate this 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 started to compete with the Romans.
Phoenicians in the period from 1200 (dates, as on most pages of this site, are B.C./B.C.E.) until the conquest by Alexander the Great in 333, lived along the Levantine coastline (and so, they would be considered eastern Phoenicians). 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, users of the western area until the Carthaginians came to power (mid-6th century).
The term Phoenicio-Punic is sometimes used for the areas of Spain, Malta, Sicily, Sardinia, and Italy, where there was a Phoenician presence (this would be the western Phoenicians). Carthaginian is used specifically 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.
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').