I’ve studied Punic civilization quite a lot, and for obvious reasons Carthage predominates the scholarship massively. But while I know that Gadir is older and had a very important temple to Melqart, and that Utica was the second largest Phoenician/Punic settlement in North Africa (I believe), I don’t really know much else about them or other non-Carthaginian Punic settlements. What do we know of their governments, people, traditions, economies, and the like?
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Our information on the "inner workings" of Carthage remains patchy at best, and we unfortunately know even less about the other Phoenician-Punic settlements situated throughout the Western Mediterranean.
Scholars have attempted to use archaeological data to draw inferences about local social and economic conditions, but this usually leads to rather general conclusions about the existence of social hierarchies (judging from the varying qualities of burials or grave goods), how the populace subsisted (based on evidence for agriculture and fishing), and what they produced and traded (as reflected in the material record). Sometimes, these data suggest contacts and even cohabitation with indigenous populations.
Religiously, the local pantheons do not appear to have differed much from one at Carthage apart from the occasional inclusion of local deities (such as the enigmatic Hoter Miskar at Maktar), and many communities evidently carried on the practice of ritual infanticide well into the Roman period. As you already alluded to, Greek and Roman sources also preserve mythological accounts about the role of Herakles in the founding of Gadir, which could derive from a genuine Phoenician-Punic tradition centered on the god Melqart.
The settlements may have modeled their governments on that of Carthage as well, with each having two (or rarely three) chief magistrates known as shophets along with a senate and/or assembly. However, since our evidence for local political systems comes almost entirely from the Roman period, we should acknowledge the possibility that these communities merely applied Punic terminology to otherwise Roman or Roman-influenced institutions.
For further reading, I recommend The Oxford Handbook of the Phoenician and Punic Mediterranean, ed. Carolina López-Ruiz and Brian R. Doak (Oxford: Oxford University Press, 2019), for a broad overview and The Punic Mediterranean, ed. Josephine Crawley Quinn and Nicholas C. Vella (Cambridge: Cambridge University Press, 2014), for more specific case studies.
Other works that I consulted:
Dictionnaire de la civilisation phénicienne et punique, ed. Edward Lipiński (Turnhout: Brepols, 1992); Ahmed Ferjaoui, Recherches sur les relations entre l'Orient phénicien et Carthage, Orbis Biblicus et Orientalis 124 (Fribourg: Editions universitaires Fribourg Suisse; Göttingen: Vandenhoeck & Ruprecht, 1993); Karel Jongeling, Handbook of Neo-Punic Inscriptions (Tübingen: Mohr Siebeck, 2008).