Web application can be defined as the application which is deployed in Web server and runs on web browser. It has web container. Eg : Tomcat, Java Web server.
While Enterprise application is the application which is deployed in Application server. An application server has
Web container, set up for messaging services, EJB container, etc. These are integrated programs which is used in large organizations. Eg : Bea Weblogic, IBM WebSphere, Oracle IAS, JBOSS etc.
An enterprise application contains enterprise beans(session/entity beans) and runs in a J2EE Container which provides transaction and security services to the beans and it is named as .ear file. Whereas a web application, contains servlets/jsps alone and named with an extension of .war., executes in a web-container like tomcat.
An Enterprise container which has special features like JMS,Transaction etc. These containers can also handles web applications.