That is a great question! The invention of the cotton gin had profound implications for the South in many ways, expanding its economic base of cotton, creating a plantation "culture," and contributing to the importance, expansion, and solidification of slavery, which, of course, led to the Civil War.
First, the invention of the gin, which afforded an easier extraction of seeds from the cotton, led cotton to become the South's biggest export, so its economy expanded considerably, with more and more land devoted to cotton growing, and more and more people finding it worth their while to invest in this crop. The large plantation, with acres and acres of cotton, was, if not born because of the cotton gin, more common, a sort of agricultural industrialization with its concomitant economies of scale. This was how the South made its money. Additionally, I believe this made it easier for the growth of some clothing industry in the South, since its primary material, cotton was right there.
The plantation culture was almost feudal in nature, with a kind of lord and lady of the manor, the plantation owners, who indulged in their own forms of conspicuous consumption with their wealth, and whose offspring tended to marry the offspring of other plantation owners, thus consolidating the wealth into fewer and fewer families. The lifestyle was one in which appearances mattered a great deal, often with ostentatious clothing and furnishings, and what we would now call conservative values were held dear.
All of this was based, naturally, upon the sweat of slavery, which became increasingly important as more and more cotton was grown. The gin did not solve the problem of picking the cotton, and cotton-picking is quite a labor-intensive process. (I know people who have actually done this, and it is extremely hard work!) The South was completely dependent upon this free labor as it expanded its growth of cotton and other crops. If labor would have had to be paid, the entire economy would have collapsed. Instead, slavery became central to the Southern economy.
So, had the cotton gin not been invented, it is likely that farms would have remained relatively small operations, cotton would not have become "king" in the South. With perhaps a greater diversification of agriculture, the plantation culture would not have developed, and slavery would not have had the prime importance it did. This could have led to the South focusing more on industrialized development, as the North did, and would have made it less likely that the South would have been willing to go to war over the issue of slavery. You could almost say that the cotton gin led to the Civil war, and the Civil War had such a profound impact on the United States that it is difficult to contemplate what our country would be like today had we not had that war, but it is possible that the tremendous tension between states' rights and the federal government would not be quite as extreme as it is, the South could have begun to diversify and invest more in its own infrastructure than in slaves, and there would not be the striking dichotomy between liberalism and conservatism as we presently have.