A Web Reference allows you to communicate with any service based on any technology that implements the WS-I Basic Profile 1.1, and exposes the relevant metadata as WSDL. Internally, it uses the ASMX communication stack on the client's side.
A Service Reference allows you to communicate with any service based on any technology that implements any of the many protocols supported by WCF (including but not limited to WS-I Basic Profile). Internally, it uses the WCF communication stack on the client side.
Note that both these definitions are quite wide, and both include services not written in .NET.
It is perfectly possible (though not recommended) to add a Web Reference that points to a WCF service, as long as the WCF endpoint uses basicHttpBinding
or some compatible custom variant.
It is also possible to add a Service Reference that points to an ASMX service. When writing new code, you should always use a Service Reference simply because it is more flexible and future-proof.