How are Java enterprises different from core Java?

Core Java is not an official name in Java platform (it is the name of a book); also Enterprise Java is not an official name.

However these two terms tend to refer to two distinct parts of Java: Core Java usually refers to Java SE which consists of the Java Language, the JVM and JDK (which itself contains the compiler, some tools and a pretty large library). Core Java is used for general purpose programming and almost anything written in Java is based on the Core Java.

Java enterprises refers to Java applications written for enterprises; the leading technology here is Java EE which consists of a set of APIs (EJB, JMS, JPA, JTA, JSF etc.) and the application servers (Glassfish, Tomcat, JBoss/WildFly etc.) which implement these APIs (or a part of them). But Enterprise Java is not strictly restricted to Java EE; someone using Spring Framework for example is still an Enterprise Java programmer.