To me, the biggest difference between these has to do with the way in which hiring and promotion is done. There is also a sense in which civil service is one type of public service.
Public service, to me, includes all jobs that relate to the government providing goods and services to the public. So a park maintenance worker is providing a public service as is the President.
Civil service, first of all, includes only professional positions. So the park worker would not be in civil service. A civil service job would be "white collar."
Second, the term "civil service" generally refers to a system where jobs and promotions are earned by merit. It is not civil service if jobs and promotions are given out based on patronage (on political connections to important people).
Civil service means two things, either a job or work force which one enters through an examination process which is relatively competitive.
The second definition is any group of workers which are in any government agency that is not the military.
So the main difference my lie in that first definition, since the second definition and that of "Public Service" are basically identical and refer to the same thing.
The examination process for the Civil Service is intense, take a look at the process for entering the Foreign Service (a branch of the Civil Service) which involves exams, interviews, group interviews and of course a very thorough vetting process where the FBI basically asks everyone you know if you ever did anything bad.
I think both are same.A civil servant or public servant is a civilian public sector employee working for a government department or agency.