And here's my second attempt, since on my first go-around I apparently answered the wrong question. (Thanks for the gentle nudge, /u/Iguana_on_a_stick.)
West Africa can mean different things. I initially assumed that OP meant western North Africa (and, to be fair, the Roman site at Constantine isn't exactly Mediterranean). In my readings on modern slavery, West Africa typically refers to the region from Senegal to Nigeria. But from the sense of OP's question, I think the area in question is the Magrib, which overlaps with the Western Mediterranean (most notably Tunisia and Algeria, subjects of my earlier response—and perhaps another excuse for my initial misreading of OP's question) but also extends into Morocco and the Western Sahara. These are areas which were integrated into the early Caliphates and through which trade with the rest of West Africa could be conducted.
The Romans weren't successful in establishing trade with these regions because at the time, this trade needed to be inland, whereas Roman commerce moved along maritime routes, specifically the maritime and riverine routes that brought food to Rome and supplies from the central provinces to the legions on the periphery. The Romans were not commercial traders like we understand them today. Basically, the Roman state contracted supply and taxation convoys to the senatorial elite, who used these state-subsidized convoys as transportation for their own commercial enterprises. And the Roman state allowed this because (1) the Roman 'state' was comprised and run by the people who conducted business in this way, and (2) by allowing this sub-contracting to be profitable, the Roman state ensured that people would continue to invest in its future. Win-win.
But this meant that the Roman economy was largely limited to the borders of the empire itself, along with some frontier trading that both kept frontier posts well supplied and kept frontier peoples dependent on Rome. This kind of economy had no need to integrate West Africa beyond the bread-basket regions of Tunisia and Algiers. Unless the state had a foreseeable interest in extending its own convoys further south—and there were no such foreseeable interests—there would be no profits to draw the Mediterranean-based commercial elites into this region.
Contrast the Islamic period, and the biggest difference is camels, hands down. Camels were domesticated and effectively saddled sometime between 300 and 500, or during the declining years of the Roman Empire. As long as the Roman Empire existed across the Mediterranean basin, investing in roads that linked inland sites to the Mediterranean, wheeled carts were the best option for overland transport. But as the Roman Empire fell apart and investment in the infrastructure of roads and bridges fell with it, camels became a very advantageous alternative. A merchant could maintain his own camel caravan without worrying about the future of the state and its investments in transportation infrastructure. Suddenly, the inland seas of the deserts became just as important if not moreso than the maritime seas that had joined and defined the classical Roman civilization.
The big change, however, happened around 700. For whatever reason—and scholars are just now beginning to wrestle with how widespread this change was—new trading sites began to appear from England to the Baltic to Central Asia to sub-Saharan Africa. With the collapse of the western Roman Empire, the infrastructure for interregional trade had largely disappeared, and the 600s had been a very bleak time, economically speaking, but for some reason, things began to turn around very suddenly.
This of course corresponds quite closely with the period of the Islamic Conquests. Perhaps because the early elite were from a region where the inland sea had already been long developed—the interior of Arabia had long been a crossroads of the Byzantine and Persian Empires—these elites were quick to develop and exploit the inland seas on their margins, the Sahara and the deserts of Central Asia, thus instigating the first truly global economy. (Arabic glass, for example, appears in North America, Greenland, China, Indonesia, and sub-Saharan Africa; sorry Australia :/ .)
In short, this is a really perceptive question that strikes to the heart of current research. For those interested, some further readings are below:
Anne Haour, "To the Other Shore: West African Trade Centre and the Wics," in Sauro Gelichi and Richard Hodges, eds., From One Sea to Another: Trading Places in the Early European and Mediterranean Early Middle Ages (Turnhout: Brepols, 2012), 441-56.
Chris Wickham, Framing the Early Middle Ages: Europe and the Mediterranean, 400–800 (Oxford: Oxford University Press, 2005).
Søren Sindbæk and Athena Trakadas, eds., The World in the Viking Age (Roskilde: Viking Ship Museum, 2014).
Richard W. Bulliet, The Camel and the Wheel (New York: Columbia University Press, 1990 [1975]). {aging, but still good; see also his recent history of The Wheel.}
Shipwrecked: Tang Treasures and Monsoon Winds (Washington, DC: Smithsonian Institution, 2011).
I don't necessarily think this was the case. North Africa was an integral part of the Roman Empire. The Roman Empire effectively began during the Punic Wars in which the city of Rome took over the governance of North Africa. Most notably, this meant sending tax shipments of grain to the city, which not only allowed Rome to become the biggest urban center of the Mediterranean, but it also established regular traffic across the western Mediterranean allowing commerce to flourish. As Christianity took hold of the Empire, it sunk deep roots into North Africa, which soon had more bishops than any other part of the Empire, which I see as a sign of the region's prosperity and integration with the Mediterranean as a whole (and, incidentally, of the peasants' abilities to play off one church faction against another).
Despite all obstacles, the Vandals thought that North Africa was too enticing of a target to pass up. And once the western administration had collapsed and the Vandals had occupied North Africa, the Eastern Roman Empire made North Africa a top priority. I think that Euro-centric histories often miss just how important North Africa was to the Roman Empire, focusing instead on Caesar's conquest of Gaul or the Gothic invasion of Spain, although it was really the invasion of North Africa that caused the Roman emperors to draw the line.
What characterized the Roman period, however, was north-south traffic. With the Roman conquest of North Africa, and the subsequent conquests of Iberia and Gaul, the Romans unified the western Mediterranean basin into a single economic region, and their tax shipments of grain were effectively government subsidies for large-scale commerce. What differed during the Islamic period is that east-west connections became increasingly more significant, at least through the early middle ages. (I'm not sure this holds as true during the later medieval period when the Caliphate lost control over the administration North Africa and the Umayyad emirate took hold in Iberia.)
Of course, the nature of trade also changed dramatically between classical Rome and the medieval Caliphates. Recent scholars have emphasized that what made the Roman economy so unique was the long-distance shipment of goods, most often piggy-backing on convoys for taxation or military supply. Once these systems broke down during late antiquity, the economy ground to a halt and most places regressed to local subsistence economies. The 600s seem to have been especially bad across most of Western Europe, although the Eastern Mediterranean fared a bit better.
As the Caliphate took over the Eastern Mediterranean and pushed their interests outward, they also expanded the commercial economy back into the Western Mediterranean (as well as out along the land routes of Central Asia and south into the Indian Ocean). Merchants in particular seem to have appreciated the economic possibilities of the expanding caliphate, and part of the reason why Islam has often been seen as a business-friendly religion is because early Islamic merchants made sure that Muhammad's pro-business ruling were well remembered. In the short term, however, this meant a new type of economy based more on the activities of individual merchants than on elites capitalizing on state convoys.
Further reading: The best study is Chris Wickham's Framing the Early Middle Ages (2005). It answers pretty much every corollary question you could imagine, or at least points you to where you could find an answer. However, as an encyclopedic work, some might find it a bit daunting. I'd point them to Hodges and Whitehouse's Charlemagne, Muhammad, and the Origins of Europe (1983). This text is admittedly dated, and scholars have challenged many of the individual premises within the text, especially as more and more archaeological material has come to light, but I think the general outlines still hold true enough, and I think it's a great read as well.