They are not the same thing.
Capper points us to the case of LogistiCare, a private contractor hired to arrange non-emergency rides for Medicaid patients. The various failures by LogistiCare, he concludes, provide proof that privatization, "costs a lot more money and provides a lot less service," than the government systems that are supplanted by private entities. Capper may be correct that he is looking at a case of too few services supplied at too high a price, but it is the monopoly power that is the problem, not the privatization.
Let's not get bogged down in the particulars of this case, maybe LogistiCare is totally incompetent, maybe not. What I can say with confidence is that the LogistiCare case is just another example of how competition is what is important, not privatization. That is a lesson that both the left and the right could stand to learn again.
For another example, look at the recent moves by school districts to save money on insurance, some by dropping WEA Trust and some by switching to WEA Trust. When WEA Trust had to compete, sometimes they won and sometimes they lost, but school districts were able to insure they were getting a good price. A great outcome for school districts and further proof of the power of competition to reduce costs.
Making a public monopoly private does nothing to reduce the inefficiencies associated with monopolies. Competition is the key.