Java Enterprise Edition
What you call the “Oracle API” is actually the Java Enterprise Edition (Jave EE) specification. I suggest you avoid using your misnomer.
Java EE is a vast collection of dozens of varied technologies layered on top of Java Standard Edition (Java SE). Various implementations of Java EE support different pieces, not necessarily all of them.
Subsets of technologies
The Apache Tomcat project, and similarly the Eclipse Jetty project and others, intentionally implement only these technologies:
These few APIs are just a small, but vital, subset of Java EE.
The Tomcat API you linked is specific to Tomcat’s implementation. Developers would only very rarely go through that API. Instead we stick to the interfaces published in Servlet, JSP, EL, and WebSocket specifications all published as JSRs. Sticking to the specs means your web app can be deployed on other implementations as an alternative to Tomcat should the need ever arise.

Web Profile
The Java EE Web Profile is a specification that includes Servlet and JSP APIs along with several more, but still a subset of all the possible Java EE technologies. Apache TomEE is one implementation of the Web Profile, that starts with Tomcat and adds more libraries. Another example is Glassfish, which is/was available in either a complete Java EE edition or a stripped-down Web Profile edition. See the Question, What is different about the Java EE packages? (SDK/normal vs Web Profile).
“Full” implementations
Some products implement all (or nearly so) of the Java EE specifications.
Sometimes this is described as a "full" implementation. I consider that label misleading as it implies the subset implementations are missing or lacking something needed. Quite the opposite. You should always use the leanest implementation that includes only the parts you need. More heavily laden servers take more memory, start and stop more slowly, and may cost more money. For example, I build and deploy highly interactive desktop-style web apps using only Apache Tomcat 8 with Vaadin 7 on top of Java 8 Standard Edition (SE) on Mac OS X.
Also keep in mind that many of the Java EE technologies can standalone, separate from a full Java EE implementation. So you can start with a leaner implementation and then add the libraries for just the few individual technologies you need. For example, Bean Validation can be used on Tomcat by adding the JAR file of an implementation.