I'm a British student taking my PhD in the US. The core difference is publication record: in the UK, publications while doing your PhD are simply nice to have. In the US, they are essential. This is one of the key time-stretchers (alongside having to take classes and being able to change your thesis at any time).
This roughly means that a PhD that has just graduated in the US is immediately hirable all over the world, whereas a European will not be able to take a job in the US until they've spent time building a publication portfolio after graduation.