Difference between SystemOut.log and SystemErr.log

System.out leads the output to the standard output stream
System.err leads the output to the standard error stream

The JVM logs are created by redirecting the System.out and System.err streams of the JVM to independent log files.

WebSphere Application Server writes formatted messages to the System.out stream.
In addition, applications and other code can write to these streams using the print() and println() methods defined by the streams. Some Developer Kit built-ins such as the printStackTrace() method on the Throwable class can also write to these streams.

Typically,
The System.out log is used to monitor the health of the running application server.
The System.out log can be used for problem determination, but it is recommended to use the IBM Service log and the advanced capabilities of the Log Analyzer instead.

The System.err log contains exception stack trace information that is useful when performing problem analysis.