First of all, I very much doubt the basic premise behind this question. I would argue that socioeconomic status is a major stratifying variable in the US and that race is far from irrelevant in Europe. Europe, for example, has serious issues with racism among fans at sporting events while non-white athletes, government officials, and businesspeople are typically given as much respect as whites in the public eye.
If we accept this premise, however, the difference comes largely from history. There are two aspects to this. First, race is more important in the US because of the history of slavery. There is no similar legacy in Europe that can make race such a polarizing issue there. Second, European countries have a history of class divisions that is not present in the United States. The US never had a hereditary aristocracy such as that which was present in France or England. Therefore, class does not mean as much to us as it does to many Europeans.