Short answer: More Nostromo, less Godfather, there's more political and less mafia activity among the very large Italian immigrant populations in South America. Different pressures, different opportunities, different regional origins within Italy, different politics made South America less fertile ground for the growth of Italian mafia.
Discussion:
First note that "Italian crime families" is anachronistic for the late 19th and early 20th century. The mafia in the US isn't a transplant of some existing Sicilian family organization-- the US "crime family" is a local 20th century development for the most part. In 19th century Sicily this was more about landless peasants from agricultural communities and their relations with absentee landowners, rather than a family's criminal enterprise.
The rise of the mafia in the US is associated specifically with Sicilian and (later) Neapolitan immigrants, not Italians generally. Taking just Argentina as an example, before 1900, Italian immigration is largely from Northern Italy (Parma, Veneto & Lombardy)-- immigration from southern Italy and Sicily does occur, but it's after 1900 and the Italian-Argentine population is historically much more influenced by the politics and culture of Northern Italy, than the South.
Italians become the largest country of origin in Argentina, so much so that Argentine Spanish is very distinct of the influence of the Italian language. Because Italians were so numerous in Argentina -- more than half the population is of Italian descent today (including such notable figures as Pope Francis), Italians weren't a minority.
Italian immigrants to Argentina and their children had the opportunity to participate in a wealthy and quickly growing nation on reasonably non-discriminatory terms; organized crime tends to rise among communities with less legitimate access to power and opportunity.
Giuseppe Garibaldi himself comes to the New World, first to Brazil and then to Uruguay, and drew young men to his banner in the Italian Legion (aka "Redshirts"). The politics and passions of Italian nationalism are more important to Italian populations in South America than in the US; that's where the regional origin of the Italian immigrants is important. Remember that Sicily has an ambivalent relationship with Italian unification-- Sicilians weren't consistently enthusiastic about the Risorgimento; Palermo revolts against Italy in 1866, a revolt which is put down with considerable violence, spurring Sicilian resentments. While its painting with a broad brush, one can say that the American mafia tends to be drawn from the Sicilian peasantry which is disappointed with what they view as unfulfilled promises of Unification; indeed that's one of the drivers of Sicilian migration to the US in that latter part of the 19th century.
The politics of Italians in South America were quite different from the US; much less attached to the Church, much more to Garibaldi, Italian nationalism and to Freemasonry. Italian mutual aid societies (Unione e Benevolenza for example, founded in 1869) had a powerful civic role, the place an immigrant might look to for assistance that didn't have an equivalent in North America. It was a proportionally larger community, drawn from different places in Italy, with different institutions, politics and opportunities than in the US. Italo-Argentine organizations are even today very significant constituents of Argentine politics, FEDITALIA represents Italo-Argentine interests in a way that doesn't have a similarly weighty parallel in the US.
So we don't see the kinds of Italian organized crime organizations in South America that you find in the US in the early and mid-20th century.
NB -- this is answer is mostly specific to Uruguay and Argentina, where Italians were the largest ethnicity or nearly so; I know less about Brazil, other than to say that Italians were a smaller portion of the population and that there's no obvious sign of a Cosa Nostra like mafia among the immigrants there either.
Sources
Immigrazione Italiana nell’America del Sud
Garibaldi in Uruguay: A Reputation Reconsidered
Italian Revolutionaries in the Empire of Brazil
ORGANIZING ETHNICITY: THREE EPISODES IN THE POLITICS OF ITALIAN ASSOCIATIONS IN ARGENTINA, 1947-1989
Peasants, Patrons, and Brokers in Western Sicily
Mafia and peasant rebellion as contrasting factors in Sicilian latifundism