OK, problem resolved, more or less.
Apparently the problem is indeed with Vista, something called User Access Control (aka UAC). It seems to stifle changes to files under Program Files, so that the state of the Tomcat installation was locked in as it had been when first created (although that doesn't explain why I was able to create the webapp the first time). As mentioned earlier, even a change to Tomcat's web.xml did not "take".
So, we used Tomcat from the zip distribution (as opposed to the service installer), unzipped to a location outside of Program Files, and now everything works as expected.