Well for starters, the UK has the Queen, while the USA has a presidential system. The UK's parliamentary system of governance is basically path-dependency Calvinball, while the USA is more formalized (although Trump is breaking a lot of norms.)
I've only been to England and Scotland, but have many friends who live there. It's easier to make generalizations about the UK than the USA, because of the vast population and size differences. The USA is more comparable to other continental countries like India, China, Brazil, Australia, or Russia. But language is still key.
Honestly, they're pretty similar in a lot of ways. Narcissism of the small differences plays things up. English people are much more shy than American people, but Scottish people are less shy.
Americans know how to say jalapeño in Spanish (Brits say JA-LA-PEE-NO), while Brits say croissant in French. Those are the two secondary languages taught in each country. Polish is the second most spoken language in the UK, and Poles are often demonized by UK racists in the same way Mexican Americans are; people tend to look down on low-wage laborers. Poles are treated much better in the USA, and Mexicans are treated much better in the UK. Asian usually means Eastern Asian in the USA (awkward for South Asians), but South Asian in the UK.
There's a European saying that Americans "live to work, not work to live." England is closer to the USA than Europe in this regard, and this will probably accelerate under Brexit. Scotland seems closer to the continent than England, and it is probable that they will leave the UK to join the EU. There are no serious secession movements to break up the USA. One of the key things I noticed was that EU citizens immigrated to the UK for a significant part of their life do not identify as British, while people who immigrate to the USA identify as American pretty proudly. I think this is largely due to the UK not having birthright citizenship, which has made the British identity weaker since the loss of Empire. An English guy I met in Germany once corrected me when I called him a Brit: "well, I'm English... I don't know about the whole British thing nowadays." Londoners of immigrant backgrounds from the former Empire are most likely to identify as British. Ads in London are much less racist than American ads, though. American ad executives don't like to show multiethnic couples, while UK ones do. There is a phrase that in the USA, "race was invented so there would be no class." Class and race have always been perhaps the dominant themes of the UK and USA, respectively.
The USA is much more diverse than continental Europe, but London and New York are more alike than New York and LA. But London is so expensive that you immediately understand why people in the rest of England voted for Brexit: to punish London for being full of opportunity at the expense of the rest of the country. The USA does have this dynamic, but with the coasts instead of just one city. A Tunisian friend of mine said "when I went to London, I could tell it was once the capital of an Empire, but an old empire." The USA deliberately chose to make its capital a smaller city to avoid the European model.
Food major USA cities usually tastes pretty good, while "traditional" British food was so bad they conquered the world to find other foods. The UK national dish is chicken tikki masala, which is delicious and amazing. The USA parallel is pizza (Italian pizza is quite different, look up the pizza effect) and the burrito, which was invented in Texas.
