A staple of the counterfactual history genre is the possibility of the Confederacy succeeding in breaking away from the Union. Most people focus on what sorts of military tactics or strategy would be needed for that success to happen, or follow domestic politics for a few decades. I'm going to assume that the Confederacy could indeed have broken away had things gone only slightly different, and would have survived.
Now consider the impact on WWI, and WWII if that still happens. The Confederacy would have been strongly pro-Allied--given its long-standing trade links with France and Britain and their tacit support during the Civil War (which would have been greater if not for especially fortuitous harvests).
The North would have been more split. You have a larger German-American population. On a strategic level, the country would have been sandwiched between a British Canada and British-leaning Confederate states. Whatever the Anglophile inclinations of elites, the incentives would have been to balance against neighboring Allies and embrace the Axis. Britain, after all, did burn down the US capital.
That is; America, by virtue of being undivided and secure, had the luxury of idealism in foreign policy. America viewed itself as an exceptional empire, rather than a full participant within a system of competing equals--yet this was predicated on geographical unity at home. A rupture after the Civil War would have raised the potential of bringing Europe-style state divisions, and Europe-style realpolitik to the Americas. The plausible corollary is that a European-style system of entangling alliances is not out of the question.
By this point of course, the North would have left the South far behind on industry and manpower. It's hard to say what would happen. Perhaps the North could use global war to re-unite the country by force; or else the South could hold off the North through Boer-style guerrila tactics. Either way--the US would probably not be contributing troops to Europe on net, and trans-Atlantic trade in supplies and goods would be badly strained. Especially by 1917, when Germany made peace with Bolshevik Russia and redeployed troops west; American reinforcements were the crucial ingredient stemming that tide and turning the war around. Without those critical inputs, it's easy to imagine Germany coming out ahead in WWI.
And then all of that feeds into WWII, etc. etc. A Germany without resentment would probably not have gone towards the Nazis; the Russians would presumably still go Soviet. It's tough to guage what would happen to the defeated Western Allies. An early loss of the colonies seems likely, probably with no partition in India. Their eventual status of the Western democracies would depend on the coming fight between Germany and the USSR, and they would then either be drawn into a Germany-dominated European Community, or live as Soviet client states.
All of this is speculation, but I think there's some value to this type of counterfactual thinking. To the extent that we can plausibly connect small changes in the past to enormous changes today--that highlights the precariousness of history as dependent on contingent events. Things didn't have to turn out the way they did. This is Niall Furgeson's take on counterfactual history; it's an account which prizes the importance of particular people, happenstance events, and the like.
Alternately, we may find that things end up roughly where they are now. This would be the determinist take on history, which has a long lineage going back to the materialists and Marxists. This is the harder type of counterfactual history to model. It's a lot easier to change one event and extrapolate forward; it's a lot harder to think of what the counter-moves and responses would have been from other involved people.
In particular, what you get at in this scenario is the huge importance of extra-European and extra-Eurasian powers in maintaining a balance on the continent. Powers in Europe have a natural tendency to try to seize hegemonic power, which Rome alone managed to pull off. The Hapsburgs, Bourbon French, Napoleonic French, the Second Reich, the Third Reich, and then the Communists--all saw their only hope of security through continent-wide domination. In all cases, this tendency was checked and prevented through a shrewd pattern of alliances sewn up first by Britain, and then America acting in concert with Britain.
But that type of balancing was very contingent. It required a unified Britain, secure behind a navy, being an economic powerhouse an possessing a strategic interest in European events. And then it required an America which was unified and which identified with the British containment mission. The first counterweight is pretty much there by virtue of geography. It's the second, American, counter which was much more iffy, but was also essential for a European balance of power. Without these counterweights, land wars in Europe would probably have continued until either the logic of atomic warfare kicked in, or one state dominated the rest.
As long as I'm wildly speculating anyway, I'm going to go ahead and say that this balance of power was pretty crucial to Europe's success. Being divided into numerous, non-hegemonic states propelled the incentives for innovation and growth in ways more absent in the Middle East, India, and China--all of which were for significant periods dominated by one autocratic sovereign.