Thinking Java EE is primarily about EJB and JMS is pretty old school.

These days CDI is the centrepiece of Java EE, along with services like Interceptors and Bean Validation. JPA, JSF and JAX-RS also play pretty important roles, which are all things you don't find by default in Tomcat.

Actually, nearly every time I see people using Tomcat they add many of the things mentioned above. You might as well start with TomEE then and work from there.