Main difference that I have found:
US - test for every possible thing that could be wrong with you, once everything is ruled out, tell you to rest, take pain killers, etc.
UK - don't test for anything, assume it's not serious, tell you to rest and/or take pain killers. If it gets worse, you have to go back and may eventually get sent to see a specialist or have to go to A&E.
There are pros and cons to both methods. With the US method, you can end up spending a lot of money and time unnecessarily.
With the UK method, conditions can get worse because they aren't treated early enough.
Also in the UK, doctors will only ask you about your specific complaint, while in the US, doctors will ask you how you are feeling generally, (how's your appetite, how have you been sleeping, etc.) to see if there is anything wrong with you that could be related to your complaint.
This is only my experience, though. I'm sure it varies from doctor to doctor.