Earlier quoted context omitted.
Dropwizard is a framework that delegates to Jetty the responsibility of serving requests by default, but you can host a Dropwizard app on whatever server you like. In Java terminology, "application server" has actually started to mean servers capable of the full Java EE stack, which really means EJB and JMS. Jetty is not capable of those. Tomcat from what I know is not capable of those either. Both are targeting firs…
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…
JAX-RS is the only thing OK-ish in the list you mentioned, unfortunately its design that ignored asynchronicity has been shortsighted (heard that it got fixed in 2.0, but apparently they also added junk).