“The difference”? There is no small difference; the US system is <em>profoundly </em>(extremely, deeply) different from the UK one. Actually, the US system seems to be different from most elementary and secondary education systems used around the world.
First, language: Americans don’t say <em>primary school, </em>we say <em>elementary </em>school. But more than that, we have some different ideas about what it should be like, especially in the first few years (which we call <em>grades). </em>You see, American kids that age are not really taught much information (facts, concepts, stories of how things happened) at all. This is because of some ideas from 90 years ago in the US that said teaching that stuff too early hurts the kids’ development. Children should explore and learn on their own, even make their own knowledge.
So American kids read some very boring, easy silly stories in elementary school so that they learn what US elementary teachers call “reading skills”, but they don’t start learning much knowledge in school before age 9 or 10. When that happens the knowledge they do get is based on what textbooks their district, school, and individual teacher decide to use. They are maybe taught basic arithmetic, and that’s it.
This is the case across the different states in the US. The set of ideas is almost universal, because it is what US teachers get taught in their college educations. There are some exceptions, like the Core Knowledge movement, but that has 1,200 schools. There are more than 15,000 school <em>districts </em>in the US.
A result of this is that, in the US, we don’t have a set of content exams for students leaving elementary school/entering high school, as I understand is the case in most other countries. The British call theirs the GSCEs.
These ideas never really crossed the Atlantic to the UK. So, if I understand correctly, in the UK kids get the “classical” knowledge-filled education from their first day in what the British call <em>primary school. </em>
<em>Very different. </em>