This is a difficult question since even within Europe there are big differences in religiosity; for example, Catholic Ireland is considerably more religious than either Catholic France or Catholic Spain, and Spain is more religious than its neighbor France. Still, I do believe we can partly explain why Americans are much more religious than Europeans.
I am confident about the importance of competition among denominations- also emphasized by Posner. America has several thousand denominations competing fiercely for members among each other and against Catholicism. Fierce competition always induces competitors to try to better satisfy the wants of their customers. This is why the many American denominations offer a large variety of religious services and doctrines that enable families to match their religious desires to different denominations.
Lawrence Iannaccone of Chapman University followed up Adam Smith’s observations on religious competition with an interesting empirical investigation in 1991 of the relation between competition among different religions within a country and the religiosity of its population. He finds a significant positive relation across countries between competition and religiosity, although one cannot be certain that the causation is mainly from competition to religiosity rather than from religiosity to competition.
The growth of evangelical religions in many Latin American countries is an example of causation from religiosity to competition. In 1900, evangelicals numbered about 1% of the population of Latin America, while by 2010, they reached 17% of this population-the rest being mainly Catholics. Latin American countries with the highest concentration of evangelicals are Brazil, Chile, and Guatemala. The Catholic Church in Latin America has often received both direct and indirect support from the governments of this region. As a result, many priests and bishops became more interested in politics and advocacy than in attending to the spiritual needs of their congregations.
This created a religious vacuum that provided an opportunity for evangelical Protestant denominations to try to convert individuals by offering the religious experiences they wanted, and that were less readily available in their local Catholic churches. This helps explain the growth of evangelical religions in this region, especially in recent decades.
Two other important contributions to the difference in religiosity between Americans and Europeans are social interactions with respect to religious beliefs, and the transmission of religions from parents to children. The religious beliefs of individuals are a prime example of beliefs that are very much affected by the beliefs of neighbors, relatives, and co-workers. An individual is more confident about his religious beliefs when the persons he interacts with hold similar beliefs.
The importance of such social interactions with regard to religiosity is that even small initial differences in religiosity between Europeans and Americans would get magnified over time into possibly very large ultimate differences between them in religiosity. Perhaps the initial differences were due to the greater competition among denominations in America, or to forces that we do not understand fully. The powerful reinforcement effects of social interactions among individuals with related beliefs would over time widen the religious gulf between Americans and Europeans. As a result, large differences in religiosity between Americans and Europeans would eventually emerge from relatively small starting differences.
Differences in religiosity between Americans and Europeans are also reinforced and maintained by the strong transmission of religious beliefs from parents to children. This transmission is much more powerful when both parents have similar religious beliefs. Men and women with similar religious beliefs are more likely to find and marry each other when a sufficient number of men and women in the marriage market have similar beliefs. Therefore, as the gap in religiosity between Americans and Europeans increases over time, due at least in part to social interactions, it becomes much easier for religious Americans than for religious Europeans to marry each other, and transmit their religiosity to their children. That difference in the intergenerational transmission of religious beliefs further widens the gap between the religiosity of Americans and Europeans.
In summary, I believe that greater competition among religious denominations in America than Europe, social interactions among religious beliefs, and the transmission of religions from parents to children go a long way in explaining the large differences in religious beliefs between America and Western Europe, even though America and Western Europe are similar in many other respects.