There are differences between the ESF system (British) and the American system. The main difference is that it is quite acceptable to leave school at year 16 in Britain and so there are public exams at this age. If you continue in education there are university entrance exams at age 18 too. Generally in the USA you finish high school and so the main public exams are at age 18, which are used for entering university too.
Most of the international schools in Hong Kong are moving over to the International Baccalaureate exam at age 18. This is accepted worldwide for university entrance. (Although you may need to check if USA universities need the SATs instead but most schools in Hong Kong will help children to take the SATs too.)
Thus the ESF have GCSE exams at age 16 and IB exams at age 18.
The American school will have SATs at age 18.
Generally the standard of mathematics and sciences is higher in the British and IB systems than in the USA systems. This is because the degree programmes in England (it is different in Scotland) is only three years whereas it is four years in the USA.
My son is currently studying mathematics at Sheffield University in England - he studies nothing but mathematics. There is no general studies year as there are in many USA universities. I had a friend whose child has a physics degree from a USA university and he didn't even study physics in his first year!
My younger son is currently studying for his GCSEs in May. It is a lot of pressure for a 16 year old but I think it helps the children to mature. He is so much more grown up now than just a few months ago and I believe the exams have a lot to do with this. It is not just his studying habits that have matured but his whole attitude to life.
I hope this help to explain some of the differences.