Several factors are involved, including the types of settlement and expansion that was predominant in the South vs the North.

The advantage of slave labor is that you do not need to significantly compensate laborers for their work (aside from the minimum necessary in calories and lodgings needed to survive), thus saving money. A less thought of but equally significant aspect is that if you are in a territory with relatively low population density (like the frontier), slavery ensures that labor is concentrated and nailed to a specific place. If you are surrounded by countryside and have free laborers, why would hey want to work grueling schedules in plantations for little compensation instead of just going into the deep country and tilling a little stretch of land themselves (whether bought, lent, or simply the result of squatting). Marx pointed this out in volume I of Capital, in the short chapter on colonization and the analysis still rings true. For this latter reason, few if any traveled South to pick cotton.

The financial problems of the Ante Bellum South were factors too. On top of alarming your neighbors, emancipation just wasn't a practical economic policy for many plantation owners. For one thing, a good deal of their total capital was not liquid but in the form of land and slaves. To emancipate their slaves before they had exhausted themselves in the fields would be, in a purely economic sense, throwing money away. In addition, freed slaves were unlikely to want to continue work (at least at the same levels as during slavery) on their master's plantation unless they were too old and infirm to move on to greener pastures. Finally, slave owners lived from crop to crop, spending and investing most liquid assets, which meant that they needed a heavy influx of cash regularly just to keep the wolves (creditors) at bay. To try to emancipate and thus slow production would be analogous, economically speaking, to asking them to jump off of a moving train.

Cultural norms also prolonged slavery. Cotton picking was seen as the work of slaves and thus a foul employment, below the dignity of free whites. A small farmer could conceivably grow some of his own cotton most would be dreaming of buying slaves, limiting his participation to mostly supervision, and thus joining the local elite.

Slavery was slowly dying out in the period between the American Revolution and the 1790s. The price of tobacco (crop par excellence of the colonial period) had declined so far that free labor was more and more viable. Many at the time thought that slavery would simply die out. Then the cotton gin came about in c. 1794 and combined with the seemingly bottomless demand for cheap cotton from the budding European (mostly British) textile industries it gave birth to a new demand for slaves. The Southern economy was already geared towards slavery and primed to take advantage of this demand so it is little surprised that rather than dismantle the system due to moral scruples they simply reoriented it from tobacco and other products to cotton. Thanks to Jefferson's purchase of the Louisiana territory from Napoleon in c. 1803, the size of the US more than doubled and a huge swath of new, arable land was opened up to plantation owners hungry to make their fortunes in the new Western territories.

One thing that Edward Baptist points out in Half has Never Been Told is that after slavery, despite protests to the contrary from some idealists, it was systematically proven that free men picked less cotton than slaves. The issue was that, as Baptist affirms and I agree, the plantation was not just about pure economic ownership of the labor force but also a terror regime which used fear and pain to push enslaved people to their limits in figuring out how to process as much cotton as possible. Free men will push themselves hard, especially when they need the money to eat, but won't push themselves that close to the breaking point, especially not without major economic compensation and certainly not for extended periods of time.

This is partly why while absolute slavery did indeed pass away, debt slavery (peonage) exploded and became one of the dominant economic forms of exploitation in the South. Largely illiterate and in any case disenfranchised blacks were tricked into being paid not in wages but in tokens which were only redeemable at the company (or simply the landowner's) store. Prices were such that workers were guaranteed to never get out of debt and debts were applied onto their children [may have misspoken as regards to if it was hereditary, though I recall reading it. As I stated, peonage is getting a little outside of my wheelhouse]. Thus slavery died but in a sense also evolved and kept on until the mid-20th century. As someone specialized mostly in the 19th century, I can't comment on the 20th century evolution of peonage, however.

Sources:

-Edward Baptist: The Half has Never Been Told

-Eric Williams: Capitalism and Slavery

-Stanley L. Engerman: Slavery and its Consequences for the South in the Nineteenth Century (in Vol. 2 of the Cambridge Economic History of the US).

-Karl Marx: Capital Vol. 1.

This is an empirical question, and it might be unanswerable due to data problems. Regardless, this is much more appropriate for a sub like /r/asksocialscience.