For seeing your dermatologist, private is a lot more convenient and quick to get rid of that itch.
If you have cancer or a serious disease, you go to the public one, even if you pay private. Not because of costs, but because there are better doctors in public than there are in private (strange but true).
In fact can be even the same doctors, but the public healthcare system has more money and better resources: better machines, more complete operating theaters, more blood bags available, more physicians... (you need two surgeons at least present by law in case one gets suddenly ill for example, so insurance companies hire senior physicians from the public system all the time for filling the quota)
The main goal of private companies is earning money, so they will try to cut costs. If a patient gets worse or the operation finds a problem or do not went as expected the patient will be quickly derived to the nearest public hospital (the insurance will be billed later for this).
If you have cancer or a serious disease, you go to the public one, even if you pay private. Not because of costs, but because there are better doctors in public than there are in private (strange but true).