Hello.
If we talk in terms of the hospital size then you will find government hospital much larger than the private hospital in terms area of development.
While in  private colleges you will find more personalized treatment it and services to the patient which makes it more trustworthy for the people, if talking in terms of salary for a doctor then the public/government hospitals are funded by the government of India that means doctor in public hospital get higher salary than the  doctor in a private hospital.
But more precisely we see so specialist doctor generally go for the public hospital while in the private hospital it is formerly known for earning money from the patient.
Best of luck.