They were not colonized similarly, well, at least they were not administered in the exact same way. Let's go country by country; we'll mostly bracket the Caribbean and then deal Spanish colonies last.
The United States came out of British colonial North America. This united more or less all the English speaking British colonies on the North American continent that had been under British rule before 1763. Though all original thirteen colonies were on the Atlantic, the country eventually extended down to the part of of the Gulf of Mexico and also all the way across to the Pacific. Alaska was purchased from the Russians shortly after the Civil War.
What's now Canada were all British colonies after 1763, but before that it was French. Quebec's separate history and legal status, and particularly the French language, civil law, and Catholicism, led it to decline to join the United States to the south. The very simplified version is that, under British rule, gradually, English speaking upper Canada (roughly Ontario) became as important as French speaking lower Canada (roughly Quebec), and the two were combined into a dominion along with outlying smaller colonies (New Brunswick and Nova Scotia). The Dominion, like its southern neighbor, gradually extended its frontier all the way to the Pacific.
There's also Belize, which was a British colony in Central America; Saint Pierre and Miquelon, which is still French territory; and the Caribbean island, which on the bigger island was basically one country per colonizer per island, and though some of the areas with several small islands combined (Bahamas, Trinidad and Tobago), I believe always in accordance with pre‐independence administrative boundaries. Obviously, there are several places Caribbean islands and island chains that are not independent (Puerto Rico, the British Virgin Islands, Martinque, Guadeloupe, etc.)
In South America, the main non‐Spanish colonizer was Portugal, whose territories remained undivided and form what is now Brazil. It did, however, expand slightly at the expense of some of the Pacific Spanish countries.
There are those triplets on the north coast, the Guianas: the former British colony of Guyana, the former Dutch colony of Suriname, and the current French overseas department of French Guiana. All three colonial powers began settlement in the seventeenth century and then just sort of stayed that way, though the British didn't directly control much land until the Dutch ceded them half the land in 1814. I don't think borders changed after that. I don't know very much about them. There were brief attempts at dinky colonies in continental North America (Sweden, the Netherlands, Scotland, Russia at least), but none of them lasted centuries so none of them left a trace on our maps.
So, now we get to the question, why does it seem like the Spanish countries divided differently in North and South America? They didn't, really. So, quick Spanish colonial history, all the continental Spanish speaking countries got their independence in a very short window, between 1810 and 1830. What happened? The shortest story is, Napoleon conquers Spain, this creates this huge legitimacy crisis in the colonies, and they start declaring independence. There's decades of war not just between the Spanish and the American‐born creoles, but also between Royalists and pro‐independence forces. In the end, the Spanish and the Royalists lose everywhere.
How did Spanish America get its borders, though? To understand that, we need to understand a little about the administrative structure of Bourbon Spanish America. At the highest level of organization, there were four viceroys: New Spain, mostly modern Mexico, the Western U.S., and Central America; New Granada, mostly modern Colombia, Ecuador, Panama, and Venezuela; Peru, mostly modern Peru; and Rio de la Plata, mostly modern Argentina, Bolivia, Paraguay and Uruguay. These were basically separate colonies, and could not trade with each other. All trade, in theory at least, had to go through Spain, e.g. Peru and New Spain could not trade with each other. It is not surprising that they would all break into separate countries, as they by design had limited connections with each other.
Below the Viceroy, was the Captaincy General. These were military governors nominally connected to viceroys, but in practice separate from them, having totally separate bureaucracies. Chile, Guatemala (all the modern Spanish speaking Central American countries), and Venezuela were all separate captaincy generals, and so it makes sense that these, too became independent. There were two others, the Yucatan and one in northern Mexico, but Mexico basically conquered and incorporated these in the first years of independence, as it tried and failed to do with the Captaincy General of Guatemala. Guatemala, due to particularly strong local politics, broke apart into separate countries at the lowest level of colonial administration (the tax district, called an intendencia).
Now, basically, that leaves us with only two questions: why did Rio de la Plata and New Granada split up differently? In the end, these countries split up at the audiencia level. Audiencias were technically just royal appellate courts districts (notice it's a cognate with audience), but they fulfilled a variety of government functions, and I have seen them described as de facto parliaments. While the densely settled and densely administered Peru and New Spain split at the Viceroy level, the more thinly settled New Granada and Rio de la Plata split at the audiencia level. In New Granada, the separate audiencias in Bogota and Quito formed the core of Colombia and Ecuador, with the Captaincy General in Caracas becoming Venezuala. Rio de la Plata is more complicated, with the bulk of the Viceroy becoming Argentina. Uruguay's independence story is complex, but it was split off as a sort of buffer state between two powerful neighbors, think a South American Belgium, under a strong British influence. Bolivia was the part of Rio de la Plata that was under the administration of the audiencia of Charcas (present day Sucre). Paraguay is weird, like I think Paraguay's history is the weirdest history I've ever read. Its colonial history starts out as just Jesuits and indigenous people, no colonist, and the Jesuits try to build a strange utopia, until they are suppressed by the Church worldwide due to unrelated politics, at which point Paraguay reverts back to a normal colonial administration. It was an independent intendencia during the Wars of Independence, and I believe most people were set to become part of the United Colonies of Rio de la Plata (Argentina), until one guy, José Gaspar Rodríguez de Francia led the charge to become fully independent. Fascinating dude, he, like the Jesuits, tried to run Paraguay as isolated utopia, though more under the influence of Rousseau than Francis Xavier. Within South America, several wars shifted the borders, but the cores remained the same as independence, which meant that the cores were largely based on pre‐independence divisions.
But the basic, tl;dr is that post‐independence borders followed colonial administrative borders. For the non‐Spanish countries, post‐colonial administration was heavily influenced by pre‐colonial administration, meaning that this basically all the land a European power had in the region (except in the Caribbean, where it broke up more by island and noting that in terms of sovereignty, Quebec was British at American independence, but administratively it was still rather French). For the continental Spanish colonies, they broke up along Viceroy/Captaincy General lines, except in the two sparsely administered Viceroys, which broke up along the lower, audiencia levels, with Paraguay and Central America being exceptions cases. In North America, the minor powers were expelled from the continent, while in South America, they stayed in Guianas. So while some of the weird things that happened were contingent, like the U.S. being able to buy Russia out, for the most part post‐colonial administration was closely linked to colonial administration.
If there's one big difference, it's that South America only really had a tiny frontier (Patagonia, with maybe the Amazon being an unofficial frontier‐‐that's certainly how Brazil treated it), while the amount of unclaimed/undeveloped/unadministered land in North America was huge. If the current U.S. and Canada had all been colonized before independence, or if Spanish independence happened after the development of administrative/communications technologies like the train, the telegraph, and steamship, and because of that, either had been organized as a colonial realm differently, there might have been a different count of countries in the two continents.
edit: Clarified some language. Also, I forgot add that Panama was part of Colombia, not part of the intendencia of Guatemala. While there were several movements to secede before this, independence only happened in 1903... when the U.S. wanted to build a canal and the Colombian senate declined to ratify the Hay–Herrán Treaty. The U.S. threw their weight behind the secessionists, and began work on the canal the next year.
North America has 23 countries, it just has three big ones. And it's not like South America is lacking in countries with a large land mass (Brazil is 5th largest in the world and Argentina is 8th, both substantially larger than Mexico).
The main difference is the Spanish colonies. America broke off in revolt about a century before Canada (kinda) became its own thing. So the only reason there are two mainland British (cough ignoring Belize) colonies is because of the history of the region (separate independence times).
In South America, the Portuguese colony of Brazil was also able to stay together. All those varied countries of South America are from the Spanish colonies (map: http://wpcontent.answcdn.com/wikipedia/commons/thumb/8/8b/Languages_of_the_Americas.PNG/220px-Languages_of_the_Americas.PNG ). So this really is a question about what forces caused the Spanish colonies to stay disunified?
Someone more qualified can give you a more detailed answer, but I'd like to point out how mountainous it is along the west coast (the main colonial population centers) and the sheer distance involved with early 19th century technology (6000 miles from Buenos Aires to what would eventually become the border of California (less than 1% of Mexicans lived in the land the US annexed) and 7700 from Tierra del Fuego to the north of the currently claimed territory).
Also, the South Americans had much less used mechanisms for self-rule. The United States benefited immensely from having long established and respected democratic traditions. This instability led to more infighting and more divisions among the newly freed colonies.
This is less a geological split and more a progenitive split. The fact that they're geologically grouped is simply coincidental and due to ease of colonizing abutting land. This can be seen further when you start looking at colonization in the rest of the World as well, such as Oceania and Africa.
The main colonizing forces in the New World were:
By far:
Followed by:
Spain and Portugal saw the Americas, and it's people, largely as a resource to be utilized to add wealth and resources to their sovereign territories (e.g. their home nation). They mainly wanted raw resources (Sugar, gold and other precious ore, etc). To this effect, the natives were largely subjugated and used for the majority of labor and, beyond religious conversion and expansion, infrastructure was minimally improved. This left the people of the land more widespread and "in charge" of their land. Once the yoke of dominion was removed, they felt no real united factors.
The English/British, French and Dutch on the other hand, saw colonization as an extension of their sovereignty beyond their natural borders. (The reason why places like French Algeria had French citizens or why Aruba, Curaçao, and Sint Maarten are a part of the Kingdom of Netherlands or Bonaire is a province of the nation of the Netherlands). To this effect, colonists tended to come from their home nation and settle and created industries which were focused on production. Items like tobacco, woven goods, etc. They created new cities and lives and brought infrastructure as well as enlightenment ideals with them. In addition, generally the colonists would be viewed as "Americans", "Quebecois", etc but they were still citizens of England or France. They also preferred to displace the natives as opposed to trying to convert or integrate them into the labor force. For this reason, these colonies had much more in common with their national kin and it didn't take long for ideas such as Manifest Destiny to form.
Alan Taylor - American Colonies, The Settling of North America
William Maltby - The Rise and Fall of the Spanish Empire
Mary Elizabeth Perry - Cultural Encounters: The Impact of the Inquisition in Spain and the New World
That all being said, it's important to remember that originally the British colonies were disunited States, many with little in common (for example colonial Georgia compared to colonial Pennsylvania). Even the original union was a weak Confederacy as opposed to it's current Federation and Canada was fairly disparate until the mid-1800's with the confederation.
hi! you may be interested in this section of the FAQ:
and a few related posts
One thing no one touched on, but seems important, is that the population is kind of different. The British didn't treat the native populations as subjects of the crown. That left the colonial governments to negotiate for land and remove the natives when "necessary" for the European expansions. While in the Spanish and Portuguese colonies the natives were considered subjects and the crown had an obligation for their welfare. So, although the Spanish weren't exactly kind, the priests did try to sort of racistly and patronizingly protect the native populations. Intermarriage also seems more common in the Spanish colonies, giving rise to the casta system.
So, my piggy back question is, assuming that I understand the above correctly, how much did existing populations' precolonial history play into the divisions in New Spain?
Also, for OP, Francis Fukiyama's last book talked a little about government institutions, specifically banks, that might have caused some of the differences in political stability between north and south America.
