Public and private healthcare does indeed co-exist in the UK
In fact, the right to continue private practice in parallel with NHS work was a compromise that finally brought the medical profession around to accepting the NHS when it was introduced in the 1940s, despite the initial fears that the NHS would bring in "socialism" and "Government control" (sound familiar!?)
http://news.bbc.co.uk/1/hi/health/7405526.stmIn general, I would say that the average person's perception of the difference between the two is that if you go privately you will probably be treated more quickly and stay in your own room, rather than on a ward.
Due to various conditions that members of my family have experienced over the last few years, I think this perception is not always accurate. Many NHS hospitals have facilities that rival private facilities. Sometimes, patients can be seen very quickly, particularly when there is a cancellation.
The system isn't perfect of course. There are periodic media reports of people being left for hours on trolleys, and patients contracting viruses like MRSA from poor cleaning standards - although you could argue that the latter is a direct result of the privatisation of hospital cleaning services that started during Thatcher's era.
But on balance, even though there are long waiting lists for some treatments, most people take it for granted that they will get good treatment that will cure them. The whole idea of whether that treatment will ruin you financially just does not come into the equation at all! Some of the scare stories that the US health insurance companies are disseminating through the wingnuts are truly weird and offensive, and really have no basis in reality - e.g. old people being euthanised on the whim of a committee, or Palin's latest suggestion that Downs syndrome babies would be disposed of by "death panels" - Just how lower can these monsters go?
The one area of The NHS that has started to take retrograde steps in recent years is dentistry. In some areas it is now very difficult to find a dentist that offers NHS treatment. The dental insurance plans seem to be far more lucrative option for practitioners.