Java EE is an standard, official, specification for a full featured Enterprise Application Framework stack. Includes stuff like Object-Relational Mapping, Security, Web Applications, database connectivity, transactions…
On top of Java EE specifications there are JavaEE implementations/application servers like: JBoss, Glassfish, WebSphere, Weblogic.
Spring on the other hand, is a framework doing lots of the stuff on the Java EE specifications, but in its own form. They don’t follow Java EE specifications and APIs for that. But they do include a Web Framework, transaction management, security and several other solutions Java EE offers.