In terms of what it offers, I think the answer is compatibility. The ASMX services were pretty Microsofty. Not to say that they didn't try to be compatible with other consumers; but the model wasn't made to fit much besides ASP.NET web pages and some other custom Microsoft consumers. Whereas WCF, because of its architecture, allows your service to have very open-standard--based endpoints, e.g. REST, JSON, etc. in addition to the usual SOAP. Other people will probably have a much easier time consuming your WCF service than your ASMX one.
(This is all basically inferred from comparative MSDN reading, so someone who knows more should feel free to correct me.)
