I've lived in the US, France, and Ireland. In France and Ireland, you see little shops on the street all the time where you can bet on sporting events, and sports gambling is a huge part of sporting culture, while in the US, though people still do bet on sporting events, it's not as pervasive.
I know that many forms of gambling are common in both places, but I'm wondering what caused the difference.
US Gambling laws are extremely different based on state/county/parish/city regulations. I don't know enough about European gambling to compare, but there's some good literature on the evolution of gambling in the United States.
I'd recommend checking out Henry Chafetz's Play the Devil: A History of Gambling in the United States from 1492 to 1955 which is one of the more comprehensive studies of the long American history. Now, some of the biggest gambling spots in the US are larger (more financially successful) casinos, but that wasn't always the case. If I understand what you mean by a betting shop, that was pretty much the case of 19th-Century US gambling, which you can find described a bit more specifically in Ann Fabian's Card Sharps and Bucket Shops: Gambling in Nineteenth-Century America.
If you're interested in just how much the state politics gets involved, I'd recommend Michael Nelson and John Mason's How the South Joined the Gambling Nation: The Politics of State Policy Innovation but that book is a bit more politically technical and much more recent (1970s-onward) than my interests in gambling.
And if you like sensationalist, Herodotus-style "I'll write what I want if I heard someone say it once," you can't go wrong with Herbert Asbury's Sucker's Progress: An Informal History of Gambling in America.