The major difference between them are:
Web Services: Its more towards the internet side. When we talk about web services it could mean from the java side (deployed on Apache) or Windows side (deployed on IIS). Testing web services is totally a different topic here.
Client Server: The system here involves a client system or a GUI (wherein a user see the front end by which he can input to the system) and a Server ( a backend usually) where in the data gets saved via the GUI.