Specifically interested in the difference between the people of Carthage and of Rome, before the punic wars were over
Unfortunately I can't give you a very full answer at the moment as I am away from my computer for today, but I can write something more detailed tomorrow. Until then, I would recommend this website: http://www2.rgzm.de/transformation/home/FramesUK.cfm
Along the side you can see there is a section dedicated to costume. It does not deal with North Africa but many of the same issues apply. One of the major problems is that we have only very scattered information on what "the man on the street" wore, as clothing does not preserve, and so we are dependent on portraits. As is the case today, clothing in a portrait does not necessarily reflect everyday ware.