I downloaded Apache Tomcat 6.0.2 And created a new server in Eclipse
From the top of my head, I thought Eclipse started the tomcat server without anything in it, i.e. no web applications. You have to 'run' or actually deploy something in that Eclipse Tomcat server so you won't get the 404s.
The fact that you do get 404 error messages indicates that Tomcat actually IS running. If you shut it down from Eclipse, you won't get those 404's anymore ;)