When you think of the largest and the most economically prosperous cities in the south, you think of Atlanta, Charlotte, Houston, Miami. However, these are different than what people typically think of as the historically prosperous cities like Richmond, Charleston, New Orleans, Savannah, etc.
However, in the north, the largest and the most prosperous are the likes of New York, Philadelphia, Boston, Baltimore, etc which are also the historically prosperous and large cities.
What caused the development of these new cities in The South? And what caused the relative stagnation of the historically large cities in The South?
I can't speak on the others, but Charleston's growth stagnated for two reasons, the establishment of Columbia as the new state capital and self-imposed limitations.
Columbia was established in 1786 and is one of the first planned cities in the US. Columbia is located in the middle of the state and was a demand by "upstate" legislators to have the capital somewhere other than Charleston to protect against a perceived favoritism to the "lowcountry" planter class. While Columbia is not a big city either, it does house state government and the University of South Carolina (and industry associated with the university) that otherwise would have been in Charleston. For more on the state politics of SC (and to an extent the rest of the deep South) Walter Edgar's South Carolina: A History is one of the best resources.
Charleston also self-imposed limits on itself, both in terms of land area and heights of building. For years (centuries really) Charleston had a law limiting the height of building's in the city's limit to 55 feet (the height of the tallest church). While this recently changed to 85 feet in 2014, there is still very much a culture of closeness with history and a general disdain for downtown industry.