Both the public and private sectors are hugh and covers a vast spectrum of activity.
I honestly dont think there is much to be gained by comparing such large amorphous categories.
The Civil Service (government departments) is part of the larger Public Service.
The Public Service comprises the Civil Service plus guards, the defence forces, local authorities, prison service, education (teachers), certain non commercial state agencies (eg Fas) and the HSE (nurses, doctors, administrators etc).
The Public Sector comprises the Public Service plus the commercial state agencies such as the ESB, Bord Gas etc.
Some public sector jobs do not exist in the private sector Guards, Defence Forces. Teachers and Nurses work mostly in the public service and where they work in the private sector their terms and conditions are largely dictated by the public sector. Some professions (Engineers, Accountants, IT professionals) are largely in the private sector and their terms and conditions are largely dictated by the private sector.