Many British people don't think about clothes very much. In Britain, as well as in the USA, men in offices usually wear suits and ties and women wear dresses or skirts (not trousers). Doctors, lawyers and business people wear quite formal clothes.
And in some hotels and restaurants men have to wear ties and women wear smart dresses. Jeans and open shirts are sometimes not allowed. It is difficult to say exactly what people wear in Britain and the States because everyone is different. It you are not sure what to wear watch what other people do and then do the same. Or ask the advice of a friend or your host. You'll fell relaxed if you don't look too different from everyone else.
British people just like to be comfortable. When they go out to enjoy themselves, they can wear almost everything. At theaters, cinemas and concerts you can put on what you like - from elegant suits and dresses to jeans and sweaters. In many ways, Americans are more relaxed than British people, but they are more careful with their clothes. At home or on holiday most Americans wear informal or sporty clothes. But when they go out in the evening, they like to look elegant. In good hotels and restaurants men have to wear jackets and ties and women wear pretty clothes and smart hairstyles. But these days most people in Britain and the USA do not wear very formal clothes. But sometimes it is important to wear the right thing.