I think the central difference is that in much of Europe, the the ideal of the perfect police officer would be someone who actively helps people to stay out of trouble, e.g. calming down a brewing pub fight situation before it gets out of hand, or warn you before doing something wrong. I guess this ideal also exists in the US, but there it is counteracted by the opposite ideal in which the police is no position to bother anyone who isn't clearly and undeniably in the wrong, but then cracking down hard. "Don't mess with Freedom unless it's of gunpoint-grade importance."
>>usrusr+(OP)
If you've never seen it, the movie "Hot Fuzz" is a brilliant piece of satire because it starts as a British cop drama and ends as an American cop drama.