"For most Americans, there's no distinction between British accents. For us, there's just one sort of British accent, and it's better than any American accent - more educated, more genteel," says Rosina Lippi-Green, a US academic and author of English with an Accent: Language, Ideology and Discrimination in the United States.
"There was a sitcom called Dead Like Me with a Brit [Callum Blue] in it. He was a scruffy, 20-something drug dealer. Even he had that sort of patina - his was not an RP accent, it was a working class London accent."
Katharine Jones, author of Accent of Privilege: English Identities and Anglophilia in the US, says the "educated and cultured" associations have a long history. "British etiquette books have been used for years; and although Americans say they have no class system, they do - and the American upper class apes the British upper class."Another point the article makes: British expatriates in Australia (where their accent is associated with complaining and being bad at cricket, and/or where refinement and intelligence have traditionally been associated with weakness and/or metaphorical or literal homosexuality rather than any positive attributes) tend to lose their accents pretty quickly, whereas those in the US (where their accents make them appear intelligent and sophisticated, and often get them preferential treatment) retain theirs. Funny, that.
America has the most obvious class system I've seen anywhere in the world. It's in your face wherever you go. If you see an American declaiming Old Europe's hereditary system, he/she doth protest too much.
America still has the myth of social mobility ("any man can become President"), and the belief that one makes one's own success/failure (hence the lack of a welfare system, as that would be rewarding laziness and incompetence). Though recent studies have shown that there is actually less social mobility in the US than in Europe (I imagine the UK would be somewhere between the two).
There was a great episode of the US show Trading Spouses that featured an uptight American family that dressed nicely for dinner because the Mum was an etiquette coach. They swapped with an English family who were rich and smart (the Dad is a Rocket Scientist) who preferred to go bike riding or paintballing than seeing the opera. Hillarity ensues.