I'm not sure what you have in mind regarding the differences, but in general it is an easily-demonstrated historical trend that the deployment of racial hierarchies and taxonomies is always a reflection of the power conditions that matter most to the people who are deploying them. So 19th- and early 20th-century eugenics is an easy case in point. British eugenicists like Francis Galton were almost totally consumed with questions of class, and when they dipped into racial issues it tended to be how to classify the Irish. These obviously reflect upper-class British concerns. American eugenicists who were based in New York City during the large waves of immigration from Eastern and Southern Europe (like Madison Grant) were obsessed with differentiating between Northern Europeans (e.g. "Nordics") and Slavs, Jews, and Italians (and, of course, the Irish). Eugenicists in the American South (like Walter Plecker) were obsessed instead with "miscegenation" between Blacks and whites. After the Great Migration (in which millions of African-Americans moved from the South to the North and West), American eugenicists (like Lothrop Stoddard) and racial attitudes in general transformed into something more about Black versus White rather than the "divide up the whites" strategy preferred by a previous generation. There are more fine-grained details one can get into regarding specific locations and their preoccupations — at various times, concern has focused on how to "properly" consider Native Americans, those of Meso- and South American ancestry, immigrants from various countries in Asia, and so on.

So without wanting to comment much contemporaneously (you'd want a sociologist for that), it's clear that the situation with race and power in Europe today is still quite different than the situation in the United States. In the United States, the history of immigration has been decidedly different, as has the history of slavery, than from Europe; the European history of colonization, in turn, is very different from that of the US. So starting with the idea that racial hierarchies and taxonomies are a reflections of these kinds of differences, it is not in the slightest surprising that these two regions (and no doubt countries within Europe) have very different attitudes. I would also just note that as in the US case, these shifts can take place relatively quickly — in the US case, the predominant racial theories (not just by experts, but "folk" ones) shifted within a generation or two as conditions changed (e.g., the Great Migration).

On the changing trends of racial theory in the United States, Matthew Press Guterl's The Color of Race in America is a good place to start.