Web systems are one type of client/server. The client is the browser, the server is whatever is on the back end (database, proxy, mirror, etc). This differs from so-called “traditional” client/server in a few ways but both systems are a type of client/server. There is a certain client that connects via some protocol with a server (or set of servers).
Also understand that in a strict difference based on how the question is worded, “testing a Web server” specifically is simply testing the functionality and performance of the Web server itself. (For example, I might test if HTTP Keep-Alives are enabled and if that works. Or I might test if the logging feature is working. Or I might test certain filters, like ISAPI. Or I might test some general characteristics such as the load the server can take.) In the case of “client server testing”, as you have worded it, you might be doing the same general things to some other type of server, such as a database server. Also note that you can be testing the server directly, in some cases, and other times you can be testing it via the interaction of a client.
You can also test connectivity in both. (Anytime you have a client and a server there has to be connectivity between them or the system would be less than useful so far as I can see.) In the Web you are looking at HTTP protocols and perhaps FTP depending upon your site and if your server is configured for FTP connections as well as general TCP/IP concerns. In a “traditional” client/server you may be looking at sockets, Telnet, NNTP, etc.