Why is WCF so important and in what cases is it used?

There's little to add to the responses so far, especially the one from "siz".

One thing to add is that WCF is the current way to do web services on the .NET platform. It's not the "new" way, it's the current way. ASMX web services are the old and just barely maintained way. One Microsoft employee has publicly stated that only critical security fixes will be made to the ASMX platform, so if you intend for your services to be useful more than a year from now, don't use ASMX.

In addition to the typical "web service" use cases, WCF handles atypical cases, like binary communication over named pipes, message queues, etc. To a very large extent, the service you write to support something simple like SOAP over SSL can also support these other protocols, with no changes to the code.