Maven project runs in IntelliJ IDEA, but mapping fails with tomcat deployment

My Maven project in IntelliJ IDEA which runs correctly from the Run->run menu. I can open a web browser to http://localhost:8080 and the app and all it's links work flawlessly. This is running Tomcat in the background to serve the servlets.

But if I click on Build->BuildArtifacts to create a war file, then copy the war file to the appropriate Tomcat directy, and start Tomcat, the app comes up but the links are all broken. I assume this is because there is a mapping difference between the way the app is run from IntelliJ and when it is run directly from Tomcat, right? I've tried manually tweaking the web.xml config file to get the mappings correct but to no avail.

Is there a way to make IntelliJ automagically create all the proper mapping config files (web.xml, etc.) so that when deployed as a war file the mappings work just like they do when launched from the Run tab?