There are several key differences between the two main families featured in My Ántonia. The Burdens are the family of Jim Burden, the orphaned boy who's the novel's narrator and protagonist, while the Shimerdas are the family of the titular character, Ántonia. And one important difference to mention (though it doesn't necessarily fit into the following list of cultural differences) is that Josiah and Emmaline Burden are not Jim's parents: they're his grandparents, so they're a generation older than the Shimerdas.
Back to the key differences. The first pertains to nationality and ethnic heritage. The Shimerdas are Bohemian immigrants (and Mr. Shimerda, Ántonia's father, feels very out of place in his new home, which leads to depression), while the Burdens have long been established in their community in Nebraska.
The second difference is in the disposition or personality type of each family's patriarch. As I mentioned above, Mr. Shimerda falls into a depression due, in part, to the alienation and disorientation he feels as an immigrant. But several passages suggest that he is a depressive character to begin with, moody and prone to feeling sad. Josiah Burden, on the other hand, isn't expressive. He's practical and hardworking, and he doesn't say much.
A similar contrast could be drawn between each family's matriarch. While Mrs. Shimerda is an unhappy sort, especially after her husband's suicide, Emmaline Burden is warm and loving to her grandson, serving a maternal role for Jim.
Another key difference has to do with socioeconomic position. Partly because the Burdens are well-established in Nebraska and partly because Josiah is such a diligent worker and provider, the family is comfortable compared to the Shimerdas. The immigrant family has many children, and Mrs. Shimerda struggles to make ends meet after her depressed husband takes his own life.