For me the clear difference is that a Public Service must directly benefit citizens needs - either through services delivery or by providing a direct interface with the public via web access, et al.
Government obviously does all kinds of things it thinks are public services - but actually are self-serving internally facing functions.
These are quasi public services - only indirectly.
Of course government is very adept at this - a criteria may be that average citizens would recognize a public service without having it explained what it is and how it is benefiting them.
