In my Java application I have a window which holds a JTextArea within a JScrollPane with scrollbars policies set to AS_NEEDED.
As I run my application I see that JTextArea this way:

Why am I seeing the scrollbars with that cutaway knob (which doesn’t reflect a “standard” representation like this)?
The Layout for the frame is GridBagLayout, and I’m on Mac OS X 10.8.2, should that matter.
This is based on the Look and Feel your app is using, and the limitations of Java’s integration with the native OS layout components. The one in your screenshot looks like Nimbus.
Swing applications always custom-render the look and feel, and don’t do a very good job of using the native OS widgets everywhere. The result is that you get weird looks that might be consistent the OS only some of the time, or only with certain layout components.
Welcome to developing cross-platform desktop apps in Java. 🙁