If you need to see a consultant, it's generally easier to get a quick referral. E.g. previously waiting list to see a neurologist private was 3 months, public was 6 months. I'm not sure how these compare now. Similarly if you need an MRI, you need to go to one of the public hospitals. If you have health insurance, you can get a quicker appointment in the private sector.
I don't necessarily believe once you're seen there is a difference in the level of care, but for many conditions, early detection is important for long term prognosis.
This doesn't mean that I believe that this is the right way to go, but it's the system we have at the moment - and seems to be going more this direction.