It is this third quality that makes the Web of Science an indispensible tool: citation indexing allows the searcher to see a retrieved article in its intellectual context, linked both to antecedents (sources which it cites) AND subsequent work (sources which cite it). This is of enormous pedagogical significance, because students can quickly see that the enterprise of science really IS collaborative and interlinked, and of great use to scholars who can use it to discover connections of which they were unaware. It is especially valuable in the sciences because of the incremental nature of scientific literature, where each article specifies the previous work upon which its findings rest.
The print, DIALOG, and CD-ROM forms of Science Citation Index are familiar and famously finicky to use, but the Web of Science form is vastly superior: quick and easy to search AND cross-connected with hyperlinks that allow one to follow chains of citations easily, and offering a wonderful "related articles" feature to broaden searches (though I do wonder about the details of the algorithm that ascertains similarity).
I did most of my searching in the Science database, but also experimented with the Social Science and Humanities databases. These seem less fully interlinked (at least in part because citations are less central to many social science and humanities disciplines than they are in the sciences, and because many links are to books rather than journal articles). The Social Science database is important to me for its coverage of the psychology literature (which falls within the sciences at Washington & Lee), but for the Humanities index I have found the FirstSearch version to be quite effective.
Whether the Web of Science could substitute for other bibliographic databases is a trickier question. Certainly the ambit of ISI indexing includes most core journals in scientific disciplines, but its coverage of specialized areas is somewhat mixed. I found that the Cambridge Scientific Abstracts databases in biology were more extensive (and had excellent indexing), that most medical topics are better handled with PubMed, and that AGRICOLA has better coverage of botanical and agricultural subject matter.
I have used Web of Science extensively in a Biology Literature course and have harrassed science faculty into exploring its wonders. Faculty from the full spectrum of the sciences have been delighted with what they were able to find, and several have commented to me that they learned things about the influence of their own work that they hadn't known.
The rumor is that the price for Web of Science is astronomical. We could only afford it via some consortial arrangement, and in any case don't have the hardware necessary to run it ourselves (our UNIX machine is HP, not supported by ISI at present). I don't know how I'll be able to live without it --the DIALOG version, even with the new web interface, is far behind in features and flexibility and is only available to students and faculty in mediated form. If I knew in which direction to pray I'd do it.