If the government got $1, it would spend every cent on treating people. If the insurance company got $1, it would try to keep at least a few cents in profit and would spend slightly less than a dollar on treating people. This is why SP would be cheaper in theory if both were equally good at spending their cash. Further, the insurance company gets to keep more of their dollar the more people they can screw over by denying coverage, while the government will not turn people away.
The much bigger effect is in preventing the wasted time on paperwork from having many different insurers however. Public healthcare usually runs administration costs relating to insurance at a few percent up to ~6% while the admin costs for the private healthcare in the US are double that at ~12% and range up to 25-30% in the worst offenders.
Source (check out the graphs
here at the bottom and
here at the top)