In the second and third centuries (100s and 200s AD) the Christian population grew in Carthage. In fact, Tertullian, a great Christian apologist and writer, talked of Christians filling "every place among" the Carthaginians. It was in Carthage, with its powerful Christian presence, that a Council of Bishops was held around the turn of the third century (between the late 190s and the early 220s AD) that 70 Bishops attended. This paved the way for the important Council of Carthage, which was held about a hundred years later, at which the Christian Biblical canon was established.
Another point worth considering is how Augustine fits into this whole controversy. He stood in opposition to the Donatist. More importantly, he was a native of North Africa. So, he knew what he was talking about. Finally, there was a economic factor as well. Most of the Donatist lost their land and were in a financial pinch. This makes revolting a bit easier. W. Frend has written the best book on this topic.
I would say the most important factor would be the life of Tertullian, whose writings profoundly influenced the Latin Church. He helped articulate the idea of the Trinity, and emphasized the importance of baptism and a conversion experience, fundamental concepts to developing Christianity. It was also at Carthage that Augustine began to formulate some of his ideas, while he was teaching rhetoric.
It seems to me that Carthage is most significant as the site of early church because of its association with Cyprian and Tertullian. Both were important writers. Otherwise, the Church at Carthage does not seem significant, and was pretty far removed from the action in Europe.
Carthage was also important in the history of the church because various councils were held there, and it was at Carthage that the contents of the Bible were confirmed.
Basically, this controversy over "the lapsed" was about how militant the church and its members needed to be. This is an issue that has implications for the church's ability to spread and to recruit new members. The more that the church demands militancy and the willingness to continue to profess belief publicly (even if that means persecution) the harder it would be to attract members.
A key factor of this church was the massive persecution that its members faced under the time of Cyprian, the Bishop of Carthage. Members recanted their Christian beliefs in the face of violence, but a controversy emerged when they were allowed back into the church straight away. Cyprian advised caution, saying that they could be reinstated, but only after showing earnest repentance.
The most important issue is the Donatist controversy beginning in 312. Essentially, some of the members of the church handed over Christian books and denied that they were Christians during various persecutions. Two questions arose: (1) whether individuals should be reinstated and (2) were sacraments performed by these questionable figures valid?
The most important question theologically coming out of the Donatist controversy was that of whether a priest had to be in a state of grace for a sacrament he performed to be valid. St Augustine decisively answered in the negative in the 4th century and that has continued to be Roman doctrine. Thus, for example, if the priest who married you was latter exposed as a pedophile, this does not invalidate the marriage.