It was only a matter of time before the malice of the revolution cooled down, and the two countries could begin to identify with each other. We have similar culture, the U.S. is mostly English and western European genetically, and we speak the same language. We have so much in common, it's no wonder since WWI Great Britain and the U.S. have been close allies.
But there are rifts. Many of these rifts have faded over the last 250 years. Great Britain pretty much tossed their "riff raff" into their colonies. Anglo's in Australia were convicts, and those in a America were large made up of "surplus" population. They had little reason to be very loyal the crown.
America was also the place of functionality, they didn't have things "just because they are pretty," like the English or French. For instance they wouldn't have the traditional European gardens or hedges, they wouldn't even leave very many trees on their land, they used all the land they could for crops. This wasn't lancey dancey rich England any more, if you didn't work, you starved and you and your family died. Europeans didn't like the reality of it all, they were used to their well established farmlands and culture.
Americans were in no way rich after the revolution, America's richest had less money than many of England's (small) high middle class. And most didn't know how to maintain their money well, more than one of America's Forefathers died in relative poverty. In Europe everything was well established, and they had a very rich class.
Another part was that all Americans (who weren't lazy) had land, and this was more of a commodity in England (where they had little). Land was easy to get in America, just move west and it was really really cheap. An immigrant can work for a while, and then buy his own farm. The British envied this, of course, and it caused another rift.
Yet another rift would be the fact that the British repudiated African slavery, but gave little freedom to the common man. It was just the opposite in America, where slavery was practiced in the south, and white men had a representative government. It was very puritanical to believe in such institutions as democracy (for white men).
But as time went on British became more free for the white man, America for the black, land was eaten up in America and they started to settle into a more "lazy" -ish life style (not necessarily bad when only a little) much like the English way, and the countries were thrown together in WWI, and they've gotten along famously since then, and will continue to grow closer.
Another reason Britain might identify with the U.S. more than other European countries, is because it's an island state. It's always been more separated from Europe, and so it can identify with it's European descended cousin America, who is also separated from Europe. But this is slowly fading away, because transportation and communication lines are becoming easier to travel/use.