ASP.NET Web Services are pretty much just that. Web Services. They're SOAP/WSDL based and provide their services only to the web.
WCF Services offer a much more flexible framework. For instance, depending on how the service is defined, it can be a Web Service hosted in IIS which serialized its data via XML and uses the REST model...or it can be a Remote Windows Service that is hosted in it's own process and serializes its data via binary. All of this is achieved using the different Service/Data contracts in WCF.
In short...you can make a WCF service look almost identical to a .NET 2.0 Web Service fairly easily but, with a little work, you can do a WHOLE LOT MORE.