If you are getting just a quick flash and Tomcat does not start, probably something is wrong with an environment variable.
In general, to get
Java programs to run without a command window showing, you execute with javaw instead of java.
If you are starting Tomcat with startup.bat or catalina.bat, it looks like
you should be able to start without a window by replacing the uses of the environment variable _RUNJAVA with _RUNJAVAW everywhere it appears. See the batch file setclasspath.bat in /bin directory for where this environment variable gets set.
Bill