First please understand that Java EE is not a single specification, but a set of specifications. While Spring is a framework.
Spring has a neat integration with many Java EE specifications including EJBs.
The idea of using the specifications is to make your application portable between different implementations of this specification. Unfortunately this doesn't work very well in real life.
Now regarding Hibernate. Again: it's a framework. Java EE includes a specification called JPA. Hibernate implements JPA. You may find other implementations. So if you are using another implementation, you don't need Hibernate.
Now, you are probably want to know the bottom line. I don't really have it. If you are studying Java, the more you know, the better it will be.
If you are looking for technologies to start a new web application, I believe it's better to use Spring with a servlet container (servlet container is part of Java EE!) like Jetty or Tomcat + to use JPA (Hibernate implementation) for persistency.
