Obviously, some of this is catch-up, as the South was relatively undeveloped for a while and then various changes made it economical for companies to take advantage of the income differential. But the development literature is also clear that convergence requires meeting certain hurdles. Going back to 1948, armed with knowledge of future trends in services, manufacturing, and agriculture, I think you'd be hard pressed to imagine the South as a leading area of growth in the country, but that's in fact what has happened.
So then the question is why did this happen. Here is a long list of potential explanations, implicating among others a big push in infrastructure, human capital development (the Rockefeller-funded hookworm and malaria eradication projects were also useful), climate, and of course AC. I find these explanations slightly ad hoc and they really seem to beg the question, while also better at explaining the earlier takeoff rather than the current trajectory. My suspicion is that within a country like the U.S., state-level institutional differences explain at least some of the difference.
And there is in fact a political explanation which largely makes my point by arguing that the breakdown of the Democrat majority after 1960 spurred economic growth. While I emphasized Republicans, the point is really more about the changing electoral dynamics and the growing political power of people who do not have quite the entrenched economic interests as people elsewhere in the country (though if I read the paper right, the Republicans do have a larger impact on growth). So the argument is that the South grew because it got better political leadership which prioritized high-growth policies, though of course other things were involved.
If you look at the right of the graph, you'll see that the growth gap appears large and growing (though blue states remain richer). These are very partisan viewpoints, but they argue fairly convincingly that Southern states are doing well and Blue/Rust Belt states are not doing so well. In some sense it's no surprise that the Chicago politician sees a bad economic climate and the Arizona senator sees good fundamentals. Of course assigning causality is hard, but it's much easier on a state level--where governors can get legislation passed and attract businesses--than on a national level, where all one really has to do is not make too many mistakes and hope you get lucky.
If you really believe these results, they point to a radically different American regional differentials. This New Yorker article is an interesting view on Rust Belt/Appalachia and their future prospects. They really don't look great, and I somehow doubt that Presidential politics--which was the focus of the piece--is really going to change that. Of course, there are really two issues here. One is the increasing regulatory and business advantage certain states hold, and the other is the continued breakdown of the American family. The second is a lot more important--I know of no community in this country that doesn't prioritize stable families, education, and hard work that isn't doing pretty well for themselves--but the first is much more open to political change. But again that comes from the grassroots, and the South was lucky in that it lacked the unions and pork and could jump ahead to the jucier bits of development. It's hard to imagine this happening in many struggling states and cities, which seem to face single-party dominance and not much of a constitutency for change (more of the same?).
The overall situation here seems to be a global trend, in which the world as a whole converges with substantial hetergeneity on political and cultural grounds. India is a good comparison as a pluralist democracy with substantial political devolution of powers to states. And you see the better run states getting better while other struggle. Freer markets seem to accelerate this process; amplifying the gains while making it harder to deny people their rationalizable worth. Many people see a "race to the bottom" but optimistically, the incentives are there for everyone to govern and produce better.