They are both different and usually you will need to see them both for an effective treatment.

Psychiatrist is a medical doctor who specialises in mental health. He/she is the one who will diagnose your condition and prescribe you medication.

Psychologist is a mental health professional that provides therapy and/or counselling for your condition. They are unable to prescribe you medication.

I think the only reason private is better is that you can choose and see your own doctor everytime if that is what you want and the waiting time is very short. However, I have went with both ways and I prefer government instituition. I went to see a private psychiatrist and I hated him so much because he is the most unemphatetic and judgemental doctor ever and not to mention I paid a lot just to see him. It could be just my bad experience and luck but I feel that doctors in the private setting only cares about your money because more patients = more money and that is how they make money I guess. However, if you are doctor in the government sector, you will still get your monthly salary no matter if you see a lot of patients or little.