The main difference between doing a PhD in the US and most other countries is the coursework component. In the US, there is usually a requirement that PhD student complete at least 2 years of coursework before they start their independent research. This means that a PhD in the US will take longer to complete. For this extra time, the US system is seen to provide a better foundation for PhD students to become successful academics.
In contrast, programmes in most other countries have no coursework requirement whatsoever. The UK has introduced skills training for PhD programmes, and PhD students are usually offered the opportunity to attend seminars and courses within the department. While PhDs outside the US are typically shorter, they are also very research intensive, allowing you to stay focused on your topic.
While each programme provides students with individual supervision, the US system also draws more on a committee consisting of 3 or 4 academics. The committee approves your proposal, provides advice, and acts as examiners. Some non-US programmes will use a committee to assess your progress, but supervision is essentially left to the individual supervisor. A committee may be used in non-US programmes to assess your progress (eg an annual review). In addition, many European (including the UK) programmes used to a committee to assess your progress after their first year, before deciding if you can continue.
In the US and Continental Europe, a committee examines the PhD, usually with reference to an external examiner. In Europe, one the the committee decides that the PhD is ready for submission there is a public viva. The viva is more a ceremonial occasional to mark the acceptance of the PhD. In the UK, the PhD is examined by a viva, but this occurs behind closed doors and the student is expected to actively defend their thesis. In the UK, the PhD is examined by two examiners: one is internal to your university; and the other is an external academic chosen for their specific expertise. The Australian system does not have an oral defence at all, there are 2 or sometimes 3 external examiners and students must defend their thesis in writing.
Another major difference is funding. In the US, when you apply for position within a PhD programme, you are also usually applying for funding. In most other programmes, you may be accepted to a programme, but you will still have to find funding.
This youtube video of Matt Moran, British Council International Student of the Year 2009, provides an excellent explanation of the difference between a British and a US PhD: