I have used the tar.gz package in the past to update Tomcat.   Here's how I've performed:

1) stop the application
2) unpack the new tomcat into the server directory of the install
3) mv tomcat tomcat.old
4) mv apache-tomcat-X-Y-Z tomcat (X,Y,Z can vary)
5) remove everything under the new tomcat/webapps
6) mv the content of tomcat.old/webapps to tomcat/webapps
7) copy the server.xml and key files from old tomcat to new.
8) start it up.

I find it safer to make a new install of tomcat and re-arrange the needed data.  It's much quicker to restore to the prior configuration by simply changing the names of the tomcat directories (e.g. tomcat > tomcat.new , tomcat.old > tomcat)