Why Run Domino As A Web Server?

Workflow. You can make your Domino system support whatever content approval workflow your organization needs. With Domino Workflow (or third-party tools) you can draw the process you want graphically and have the system build the code to forward documents to approvers, send out reminders, and so on. A few other systems offer multi-author workflow, but usually you only get the workflow they've decided you want.
Internationalization. You can build proper multilingual sites which use the users' browser settings to provide the languages they want automatically, as long as you've translated the appropriate pages -- and fall back to a default language otherwise. Of course, you can build the site so different languages have different workflow, so French pages can be approved by someone in France.
Extremely rapid development. A modern web site needs approval workflow, automatic indexing/searching/expiry of content, personalization, and multilingual capabilities. A good Domino designer can have something like that up and running in days rather than months.
Flexibility. There are other systems that can produce a site as quickly, but often you're stuck to one of a selection of basic page layouts.
Decentralized and scalable. You can have servers around the world providing mirrors of the same content automatically; each server can scale from a PC all the way up to an IBM S/390 if necessary, and you're not locked in to a single OS either. Clustering allows automatic failover. Plus if the internal network links between servers fail, it doesn't matter -- the systems catch up on what they missed automatically next time the link is up.
Extremely rapid and painless evolution. You can change the design and content of a Domino web site while it's up and running. Changes to the site are automatically distributed to all your servers with no work on your part. You can turn around a site upgrade every day if you want, and have content flowing out continuously 24/7.
Design, logic and content are separated. Content authors don't need to know about programming or web design; they don't even have to know HTML. You can also change the design without having to recompile all your code, so designers don't generally need to know about programming.
A big weakness with the JSP approach is that your HTML gets buried inside your Java code, which means fixing HTML problems or changing the web site design can require major Java programming skills.
Open systems with no lock-in. You can get data in and out of Domino using SQL, COM, CORBA, ODBC, JDBC and XML. The server and client talk open standard protocols -- SMTP, HTTP, LDAP, NNTP. [My rule is never put your data into something unless you know how to get it out again.]
Offline support. You can build your web sites so that they can also be accessed and maintained via the Notes client, or using DOLS/iNotes. Then you can work on the site or write content on the plane, and have it automatically sent to the server next time you're in the office. [If you use JavaScript for things like field validation, you can use the same form designs in both the Notes client and web browsers, reducing the cost of implementation.] You can take your entire site and run it from a laptop for client demos, etc. Or you can link your site to WAP phones, PalmOS devices, ...
Security. Public key encryption to verify authorship of content, and transparent encryption of data on your laptop in case it's stolen.
It's more than a web server. It also integrates directory services, e-mail and newsgroups, and that's before you consider add-ons like QuickPlace, SameTime and fax gateways. Plus, there are the Business Partner solutions to make Domino help with everything from ISO 9000 auditing to personnel recruitment. You may not think you need a timesheet system that works across the web right now, but when you do there will be one which integrates with your Domino system.
It doesn't take over your life. You can run Domino as a web server but not use it for directory services. You can use it to run your helpdesk but not for e-mail. It's not an "all or nothing" proposition.

And to be fair, some weaknesses:
It's not a relational database. Occasionally you really do need relational operations, in which case Domino alone isn't enough. (But Domino will talk to a relational database, of course.)
The built-in web search is fine for simple keyword search, but could use more options for ordering the search results.
The HTML generation isn't perfect. To get full HTML 4.0 standards compliance, you really need to build templates and have Domino fill in the gaps. This can make dual-mode (Notes and web client) design more time-consuming. Having said all that, web design tools that will actually create standard HTML 4 + CSS are very rare.

You can find a list of impressive sites that use Domino here.