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).