The World Wide Web is the virtualization layer. All digital assets (databases, files, executables, down to the record level) can connect in a consistent, universal, two-way fashion.
The Semantic Web is the conceptual layer. The virtualized assets can be combined and linked. Those links can be used by people, machines, other (imagine emailing someone a link to a Web page). It's an important concept, because you as a person are more than what's contained in a credit card database, blog, school transcript, etc. With the SW you can link all those assets together and create a more complete picture of yourself.
In an enterprise, this could be a more complete picture of a customer, chemical compound, financial market, etc.
See the ugly picture with non-standard symbols & terminology (I'll replace it with something nicer - full size here):