The difference is that with private insurance you can switch companies and try for a better policy and better service. It is also far more likely to be efficiently run, and therefore not go out of business or suddenly cut benefits like government run programs will have to do when the money runs out. With the government you've got exactly zero choices and if you don't like what they did what are you going to do? Sue them? Good luck with that.
Zero choices, eh? OK, you want to talk about choices?? First of all, in the US most health insurance is provided by one's employer, so there is effectively very little or no choice involved, in most cases you get to take whatever your job provides, and that's that. Secondly, changing from one greedy insurance company filled with people who get bonuses for denying care to another such company gets you what, exactly? Certainly nothing in the way of increased value or better care, nothing of any substance. THERE IS NO CHOICE FOR THE CONSUMER UNDER THE CURRENT SYSTEM! There's barely even the illusion of choice.
I simply don't understand why the anti-single payer crowd can't see that there can be no true free market in health care, especially under the current system. Consumers are allowed no real choice about virtually any facet of the care they receive, in most cases it's not even possible to find out in advance what a given procedure will cost. Unless health care consumers have access to information like the true cost of medical tests and procedures, along with information about the competence and reliability of the professionals who administer them, nothing resembling a free market is possible. This is obvious, it's basic economic theory. Why is it so hard to admit that health care just might, just possibly might, be an area unsuited to a purely free market solution? Remember, emergency medical care must, by definition, be administered immediately, often without input from a potentially unconscious patient. No patient choice = no free market. It's that simple. And as we've seen, attempts to impose a pseudo-free market via private insurance companies simply leads to the mess we have now: 51st in life expectancy. For god sake, a single payer system is not just the most efficient way for a modern industrialized society to deal with health care, it's the only way! Anything else leads to a grossly unfair and unethical two or more tier system, and life expectancies comparable to third world countries. We can surely do better than that.