I assume just deinstalling tomcat6 doesn't remove the config files that are placed in
/etc/tomcat6. So you will have to purge tomcat6 using
sudo apt-get purge tomcat6 before re-installing the package. Additionally, there may be cached files in
/var/cache/tomcat6 that may affect a reinstall, so you should check that directory before reinstalling, too.
If purging tomcat's config doesn't solve the problem, there may be more causes for tomcat stopping to work, e.g. a messed-up java setup. To check this, the easiest way is to install a second JRE (OpenJDK, for example), modify the alternatives so the newly-installed JDK is used (
sudo update-alternatives --config java), and restart tomcat. If this works, the problem is likely caused by the config of the previously used JRE/JDK.
If it doesn't work, more information about tomcat's malfunction are required: details about tomcat's behaviour, error messages when running tomcat manually from the command line, contents of tomcat's logs (
catalina.out and so on).