In Lincoln's second inaugural speech, he underscores that both North and South had profited from slavery but that both sides had paid for that profit beyond measure. In what ways did each side profit from the institution of slavery, and which side profited the most? We know who paid the most dearly -- and that was the slaves and their families themselves.
The only economioc benefits of slavery were for the owners themselves, they got free labour. For the economy as a whole it was clearly negative, because by denying so much of the population the right to earn a fair salary and to spend it on what they wanted you're also hurting the growth of all the economy. And dependence on slavery led to a decrepit model of production: the enormous estates producing just one crop like tobacco or cotton, so you don't have any diversification of the economy.
I think the south profited the most. First, they held out the longest in terms of abolishing slavery. While many Northern states had freed slaves before the start of the war, many southern plantations were still striving. Second, the Southern plantation owners were more vicious and merciless forcing slaves to work and produce ridiculous amounts of crop which is why the southern economy relied so heavy on slavery.
The most obvious profit was free labor. As far as who profited the most, I don't think it was one or the other. The South was focused more on agricultural exploits while the North was leaning more towards industrialization. Each required huge amounts of manpower in those days.