The Semantic Web and the Web are basically two names for the same thing: a global information space wherein something navigates with URIs. The Semantic Web exists in the Web, and is a part of the Web at the same time. This makes them inseparable at the URI-level, and consequently, that level is not useful for explaining the Semantic Web or relating it to the Web. Analogously, you cannot compare two stories in a book by comparing the individual letters - one needs to increase the level of abstraction of concepts to make that kind of comparison. If one increases the abstraction one sees that the Semantic Web and the Web are quite different, but also shares a few concepts.
Basically, the name Semantic Web comes from that the fact that it "represents" a set of semantically and formally interlinked data units - thereby creating a semantic web inside the Web [Berners-Lee, RM]. But this should also indicate that there exist important conceptual differences between them. Roughly, there are two conceptual differences between the Semantic Web and the Web: (1) The Semantic Web is an information space in which the information is expressed in a special machine-targeted language, whereas the Web is an information space that contains information targeted at human consumption expressed in a wide range of natural languages. (2) The Semantic Web is a web of formally and semantically interlinked data, whereas the Web is a set of informally interlinked information.
By examine these conceptual differences, one finds that there are similarities; and these similarities are the use of links and their importance. It is easy to separate links and human oriented textual information; a text document basically means the same thing to the author with or without links. But the use of links can highly increase the understanding and precision of the information - to a viewer. Instead of only writing: "At the conference I met Benny", the semantics and precision would be increased to a viewer if the word Benny were a link to that persons homepage, and conference was a link to the conference. The use of links is not only to make it possible to navigate in the space, but also to share concepts.
The separation of data expressed in the machine-targeted language and semantical links on the Semantic Web is harder to separate, and there is a reason for this: The "meaning" of the information expressed in the machine-targeted language is defined by making semantical links between different concepts. The semantics of the data expressed in the machine-targeted language thereby highly depends on how its parts, or the descriptions of its parts, are semantically linked, i.e. how one concept relates to another concept. This is very similar to how humans communicate (a bit simplified). Languages used by humans express meaning by referring to a set of shared concepts that grounds the understanding of their communication. This sharing and building of concepts has taken thousands of years to establish. Two applications developed separately of each other have no shared language but possibly shared concepts. By creating universal concepts identified by URIs, we can create the set of shared concepts that machines need if the machines are going to comprehend the machine-targeted language. If a language should become universal the concepts that are used has to be universal, and by using URI they are. Hence, the use of semantical links on the Semantic Web is to share concepts.
The biggest difference is that the Web is aimed at human consumption through the use of rendering software and the Semantic Web will ONLY be used by machines. The debate on whether the Semantic Web is the "new" Web is on the wrong level of abstraction. At the URI level they are the same thing, but at a higher level they differ substantially.
[Berners-Lee, RM] Berners-Lee T, 1998: Semantic Web Road map, URL: http://www.w3.org/DesignIssues/Semantic.html