Your question covers a large sweep of history. During the time period of the 18th to 21st centuries, there was a large migration of people from Europe and parts of Asia to the New World. Immigrants in the 19th century largely came from southern Europe, including Spain, Greece, and Italy, and from eastern Europe, including Russia and the Slavic countries. Jews also immigrated in large numbers to the New World, facing anti-Semitism in Europe and Russia. They were propelled by wars and violence, including ongoing wars in Europe and civil wars and rebellions in countries such as Italy. Many faced dire poverty, limited opportunities, and persecution. These immigrants went to both North and South America, fueling urbanization and the growth of large multicultural port cities such as New York and Buenos Aires in Argentina. In addition, in the United States, there was internal migration, such as the Great Migration--the movement of African-Americans from the south to the north in the period after Reconstruction to the 1960s. They came north in great numbers during World War I and World War II in search of jobs. Similarly, in Argentina, many people went to the city of Buenos Aires in search of jobs (and this type of urbanization occurred in other cities such as Mexico City). These cities became multicultural in nature, with Spanish, Italian, Syrian, Lebanese, Greek, Jewish, Polish, Russian, Irish, English, and other types of immigrants. In addition, these cities grew in response to industrialization, as the mechanization of agriculture reduced the numbers of people needed on farms. These formerly rural people flocked to cities, swelling their populations. New industries formed in cities such as New York and Buenos Aires to attract people to work there.
Politically, both the U.S. and countries in South America such as Argentina went through revolutions to become independent from their European colonizers (Great Britain and Spain, respectively). Following independence, they both also went through civil wars (the U.S. in the 1860s over states' rights and slavery) and Argentina after independence in 1810 (over the question of whether to have a centralized government or a federation). The United States went through various periods of militarism, such as fighting the Spanish-American War, World War I, World War II, the Korean War, and Vietnam, and Argentina became a military dictatorship in the 1960s. This period lasted until 1983, when the military dictatorship was brought down in part because of the country's failed participation in the Falklands War against Great Britain in 1982. The military and political developments of the U.S. and countries such as Argentina diverged in the 20th centuries, as the U.S. became a world power, while Argentina was more isolationist. While the U.S. and countries in Latin America such as Argentina were similar in their 19th century immigration patterns, political origins, industrialization, and urbanization, their development in the 20th and 21st centuries has differed.