As you stated, JAVA_HOME points to the JDK installation path given by the Environment Variable(%JAVA_HOME%).
But java.home points to the JRE installation path, now it returns the JRE that was used to run the application, please remember that you can have multiple versions of JRE and JDK on the same server/computer
And you can run an application specifying what jre or jdk you want to use.
So, for example, if you have on your Environment path:
%JAVA_HOME% = C:\Program Files\Java\jdk1.6.0_24
But if you ran the application using an specific jre:
"C:\Program Files (x86)\Java\jre1.8.0_73\bin\java" -jar TheJavaFile.jar
Inside the application on run-time, you will get on java.home a different version of the JAVA_HOME
This may explain why on some cases you get different versions for both variable and system property.
Also, please notice that the paths may be quite different, since JRE is a different product than JDK, then they are installed in different locations, because they are independent
Now, regarding what's the difference from one JDK vs JRE, this diagram explains it pretty clear:
JDK is a superset of JRE, and contains everything that is in JRE, plus
tools such as the compilers and debuggers necessary for developing
applets and applications. JRE provides the libraries, the Java Virtual
Machine (JVM), and other components to run applets and applications
written in the Java programming language.