what is the reason for the existence of the JAVA_HOME environment variable?

Environment variables are strings that contain information such as drive, path, or file name.

The JAVA_HOME environment variable points to the directory where the Java runtime environment (JRE) is installed on your computer.

The purpose is to point to where Java is installed. $JAVA_HOME/bin/java should execute the Java runtime.

When you run a Java program you need to first start the JVM, typically this is done by running an executable, on Windows this is java.exe. You can get that in lots of ways for example just giving a full path:

C:\somedir\bin\java.exe

or may having it on your PATH.

You specify as command line arguments some class files or Jar files containing class files which are your program. But that's not enough, the java.exe itself needs various other resources, not least all the Java libraries. By setting the environment variable JAVA_HOME you specify where the JRE, and hence all those runtime resources, are to be found. You should ensure that the particular Java you execute matches the setting of JAVA_HOME.

You may also need to put JAVA_HOME\bin early on your path so that if the java.exe spawns a copy of itself (for example to compile a JSP) it picks up the right version.