Java EE Containers vs Web Containers

Java EE container: Manages the execution of EJB, JMS, JTA run on the Java EE server e.g. JBoss, Glassfish.

Web container: Manages the execution of web pages, servlets, and some EJB components for Java EE applications. Web components and their container run on the Web server such as Jetty, tomcat.

First of all, "J2EE" is an obsolete abbreviation, it is now simply called "Java Enterprise Edition" or Java EE.

Contrary to the servlet container (e.g. Tomcat), "full" Java EE application servers contain also an EJB container. EJB are Enterprise Java Beans and you can read a lot about them for example here (chapter IV). EJBs are now in version 3.2 (Java EE 7 and Java EE 8), previous versions are 3.1 (Java EE 6) and 3.0 (Java EE 5); however the greatest difference is between v2 and v3.

EJBs are designed to keep a business logic of your application. For example, stateless session bean can calculate something, or represent a Web service or whatever your application needs to do. Message-driven beans can listen on message queues, therefore they are useful if you want asynchronous communication. Singleton beans guarantee one instance per bean etc.

Regarding the file type, EJB is packed into a .jar file, Web application into a .war file, and if you want to mix them in a single application, that would be the .ear file ("enterprise archive").

Beside EJBs, "full" application server also takes care about transactions, security, JDBC resources... I would highly recommend using it over a servlet container, but the benefits come with the complexity so you will have to spend a reasonable amount of time to learn how to deal with e.g. Websphere (Payara and WildFly are much simpler, and are my favourite). JBoss and Weblogic are also quite popular, and if you are familiar with Tomcat take a look at TomEE.