What enduring impact did slavery have both in Africa and the Americas?

In the Americas, one of the legacies of slavery was continuing poverty among African-Americans. Emancipation in the United States, Caribbean, and South America didn't necessarily improve the economic lot of the newly-freed peoples, many of whom were pressed back into service by their former masters in order to prevent an economic downturn. This means that in the American South, sharecropping replaced slavery as the best means of securing cotton production, and the newly freed slaves transitioned into a kind of wage servitude that tied them to the land for decades on end.
The absence of education for slaves in the United States meant that the new freedmen effectively had to be educated en masse beginning in 1865. Reformers during Reconstruction did make attempts to correct these imbalances, though their work was hamstrung by systematic underfunding and occasional acts of terrorism aimed at white teachers willing to work with African-Americans. In any event, newly freed slaves were poorly prepared for independence.
The fact that slavery existed along racial lines in the Americas also went a long way in cementing racist attitudes toward people of African descent. Slaveholders rationalized and justified their policies in newspapers, pamphlets, and books, and much of this rested on the supposed inferiority of slaves, who were said to be dangerous and disposed to violence—particularly sexual violence. Additionally, racial hierarchies existed in the American South that placed poor whites above African-Americans. Those attitudes didn't disappear with the end of slavery and have done a great deal to fuel long-lasting bigotry and violence.
There are also certainly arguments that slavery contributes to a region's economic underdevelopment. Certainly in the Americas, slaveholding regions also became increasingly devoted to monocrop production of certain goods: cotton in the American South, sugar in Cuba and parts of Brazil. Transitioning away from just producing one kind of good proved to be very slow: abandoning the revenue from selling sugar would make diversifying one's economy expensive in the short-run. But excessive dependency on a good also meant that a sudden downturn in price could send a region's economy into free-fall.
As for Africa, the slave trade was the region's introduction to Europeans and European trade patterns. Many of the first generation of European toeholds on the continent were slave forts where traders could bargain with local chieftains, selling European goods such as firearms or iron tools in exchange for captives. As Europeans such as the British abandoned the slave trade in favor of so-called "legitimate trade," seeking palm oil, rubber, ivory, and other goods, old trading partners began to provide new goods. In short, the slave trade helped to loop parts of Africa into trading networks with Europe and the Americas. Many Africans traded not necessarily for weapons but for tools, as European iron tools could be used to farm more efficiently and to then sell food back to the Europeans (who needed food for slaves bound for the Americas). 
European trade for slaves also had a profound effect on Africa. The reality is that the African slave trade could not have been carried out without African middlemen who helped to capture and sell slaves. And while there had been forms of slavery in Africa prior to the Atlantic slave trade, the numbers of people captured in the Islamic slave trade in Africa paled in comparison to the 20 million captured for the Atlantic Slave Trade between 1500 and 1850.
Furthermore, the existence of such a large trade meant an increase in West African kingdoms, whose function was to wage war for the purpose of capturing slaves to sell to the Europeans. The Fante Confederacy, the Ashanti Empire, and other West African confederacies all relied on this trade. Unsurprisingly, when the slave trade ended, their economies were severely undermined, and they were thrust into chaos just as European states were beginning to expand territorially in Africa. The slave trade also undermined southern Africa as well, and one historian, Julian Cobbing, has argued that the Zulu state expanded southward in the 1820s to try and escape the increasing Portuguese slaving activity. This set off further war in southern Africa, and pushed the Zulus into conflict with other southern African tribes as well as the Dutch Boers and the British.