A civil servant is a person who works in the state sector, but not necessarily in a government ministry. They could work for a state-owned company as well. In English we mostly think that a civil servant works in an office, and does administrative work. We sometimes also call them "paper pushers" because it seems it is more important for them to move forms and documents off their desk to avoid responsibility.
A public servant is a member of the governing class at all levels. In the western nations we would say they are politically elected, or they are politicians. In China, a public servant would be a senior members of the Chinese Communist Party, or a member of the politburo.