This question can't really be answered as written, because traditional categories of race are somewhat arbitrary and the ones in the West are fairly fuzzy about the populations in the relevant regions now, let alone thousands of years ago. (Plus the Caucasus is a mountainous area between the Black and Caspian seas, which is surely not where any of these people came from; it just goes to show how arbitrary the category is.)

But we can interpret it instead as "Were the Carthaginians and the Romans in the same ethnic group?" I think this can be answered and the answer is no, but the details might be surprising. [Here's an interesting paper that combines history and genetics](http://www.cell.com/ajhg/fulltext/S0002-9297(08)00547-8). DNA evidence from modern people confirms the historical record: the original Carthaginians weren't closely related to the Romans but they weren't closely related to the neighboring Berbers either; instead, Carthage was a colony of the Phoenicians, who came from the eastern Mediterranean coast (modern Lebanon and Syria). If you go inland from the site of old Carthage, you find a genetically distinct population, which is descended from those who lived in the area before the Phoenicians arrived. Both are distinct from the people of the Italian peninsula.

Attempting to discuss race in an historical context is incredibly complex, as race itself is a social construct, as opposed to the immutable pillar of categorization that some people see it as. So while I can't speak specifically to the hue or tone of the Carthaginians' skin, I would suggest you check out this Monday Methods on the topic of race itself in historical context.