When you compare the U.S. with Europe it seems that the U.S. is much more religious (however you chose to measure it).
Why?
I saw a comparison a while back between the US and Denmark (one of the less religious European countries). The percentage of people who say that they believe in God is significantly higher in the US than in Denmark. But when Danes are asked if they believe in a 'higher power', the percentage saying 'yes' rises.
And I'm reminded of a close friend, an American engineer who to my knowledge never entered a church except for weddings and funerals and never picked up a Bible that I know of, once telling me that anyone who doesn't believe in God is 'stupid'. I don't believe in God, so I asked him what the word 'God' meant to him. He waved his arm to encompass everything and said, 'There has to be more to reality than this!'.
So for him, 'God' was apparently a synonym for a vague sense of transcendence, not unlike the Danes' 'higher power'. I think that there are a lot of people like my friend here in the United States. Which suggests that some of the differences in between the US and Europe might be a function of how words are defined.
I think the answer is that the state is still generally involved with religion in Europe and therefore people want no part of it.
Yes, I agree. Europe has historically had state churches, very closely associated with the ruling kings and princes. Even in relatively tolerant England, individuals who refused to join the state church ("non-conformists", that's where the word originated) were sometimes refused entry into schools and weren't allowed to hold public office. In other places, non-conformists were in danger for their lives. So non-conformity became widely associated with freedom.
Then, around the French Revolution and in the 19th century, when overthrowing "old regimes" in the name of "progressive" republicanism was a major historical force, the threatened kings found allies in their state churches in defense of the old orders. We see France swinging crazily back and forth during the 1800's between numbered republics and periods of renewed monarchy with various 'revolutions' marking the changes. This kind of turmoil and the role that the state churches played in it pretty much discredited the churches as far as more 'progressive' European opinion was concerned. There's a strong tradition of anti-clericalism in European popular culture even today that we don't see here in the United States.
Also, the Puritans were the ones who left England to come to the U.S. and those were the fundamentalists.
It wasn't just fundies. Britain's American colonies became a refuge for non-conformists of all kinds. The English king gave royal charters to the new colonies in hopes that all the damned non-conformists would move there and leave England to the Anglicans. So the Puritans moved to New England, Catholics moved to Maryland, and many Quakers ended up in the vicinity of Philadelphia (which helps explain why that city was the cultural capital of the colonies). So many Baptists moved to the US that Baptists are far more prevalent in the US today than in Europe. All kinds of unpopular groups like the Unitarians and the Jews washed up on our shores.
Here in America, churches didn't represent established old orders and the heavy hand of monarchs. Instead, people's ability to worship freely as they choose became a core principle of what American freedom meant. Belonging to one's church was seen as an expression of that freedom. This is why the American 'founding fathers' were so concerned to prevent the 'establishment' of government churches and it's why Jefferson wrote about the need for a 'wall of separation' between church and state.