As an important note, I'm familiar only with Latin American history intimately, so I can only really speak to that side of your question. The short answer is that Latin America did fracture, quite spectacularly in fact.

So while both Latin America and Africa are both made of former colonial nations, it is important to differentiate between the eras in which these two areas achieved independence. Latin America achieved independence in the age of revolutions during the beginning of the 19th century, after more than 300 years of constant Iberian presence and interaction. Like the American Revolution, the leaders of Latin America's revolutions (with perhaps a slight exception in Mexico initially) were disaffected "Americans" who were culturally very similar to their Spanish or Portuguese cousins except they happened to be born in the Western Hemisphere. This in great contrast to Africa which achieved decolonization in the wake of WWII and where with only a few exceptions there were only small populations of white settlers, and many of these fled in the wake of decolonization struggles.

Latin American natives suffered immense losses from the initial conquest, where as much as 90-95% of natives died due to Old World diseases. In the wake of this traumatic shock, most natives had no real option other than a wary accommodation with the Europeans. Many of the pre-Columbian structures of power were reoriented into imperial and religious ones. Thus Latin America circa 1800 was divided into imperial administrations governed by viceroys or intendants. The borders of these administrations were somewhat porous and at times ill-defined, but the existing structures of creole power and state authority were contained within them not across them. And within these large bodies were various other subdivisions, which, though connected with the central administration of the viceroyalty, had its own regional structure.

Now as soon as independence is declared, these administrative regions immediately begin being pulled apart by regions and sectional differences and demands. In what is today modern Colombia, for example, there is a period of history literally known as "la patria boba," or the "foolish fatherland," where tens of cities declared their own separate independences and then began fighting each other over hegemony of the old division of Cudinamarca (Colombia).

As the Latin American nations defeat the Spanish and actually begin to govern themselves, we again begin to see this sectional disintegration of the nations. To use Colombia again, it was originally a country known as Nueva Granada where it was joined with modern day Venezuela, Ecuador, and Panama. This union shattered as the country's nascent institutions were unable to manage effectively the sectional differences. These countries were actually subdivions within the viceroyalty of Nueva Granada which resented central authority resting in Sante Fe (modern Bogotá), far from their capitals, as well as perceived lack of equitable federal power distribution between the nations. Indeed, every Latin American country except Brazil was broken up after independence into the nations we know today (Panama being an exception as the product of U.S. intervention).

So to summarize, Latin America did fracture first along the largest administrative viceroyalty or intendancy, and then again into various subdivisions. This is in spite of many figures, Simón Bolívar being the most famous, who dreamed of a pan-American union. It is important to compare this to the United States' revolution and see how close they were to breaking up as well (in fact the U.S. did break up a hundred years later, like Latin America, along sectional lines). Each of these Latino regions had distinct geographies, peoples, and economies which, like the civil war United States, made compromise increasingly difficult and then untenable.

This may have been difficult to follow (and indeed, reading sources from the period are just as difficult to follow the myriad regional developments), but I hope it helps answer your question.