Problems resulting from the mixing of Swing and AWT components have been a constant source of confusion to Java newbies. This quote explains the reason for the difference very well:

Most of the issues related to mixing AWT and Swing components are related  to the mixing of so-called heavyweight and lightweight components.  A heavyweight component is one that is associated with its own  native screen resource (commonly known as a peer). A lightweight component is one that "borrows" the screen resource of an ancestor (which means it has no native resource of its own -- so it's "lighter").

From: Mixing Heavy and Light Components

The lightweight approach, taken by Swing, is the preferred, since it consumes less resources. However, there are cases where one might need to mix the two in the same application, which can result in undesired side effects. A case in point, from the same article and where all the code used below is found, goes as follows. The menu on the right behaves correctly, while the menu on the left does not:

The reason for the difference is that the badly displayed menu in the second screenshot is Swing and lightweight, while the other menu is AWT, and therefore heavyweight. The Button is an AWT component too, therefore being heavyweight. Unless one forces a lightweight component, via setDefaultLightWeightPopupEnabled(false), to behave like a heavyweight component, unwanted side effects such as the above will result.

But all that is old news, of course.

The new news is that already with the early access release of JDK 6 Update 12 one can, without needing to do anything at all, obtain the desired effect, which is as follows:

This is exactly what one would want: the rendering of a lightweight component over a heavyweight component. Though the major features of the EA release are slated as "Windows 2008 Support" and "Java Plug-In now supports 64-bit browsers", it is this heavyweight/lightweight fix that I find the most useful (since I am neither a Windows- nor a 64-bit browser-user). If one looks through the list of issues fixed in the current JDK 6 Update 12 build, one notices that a lot of fixing has been done in the Swing & AWT packages. 

The final screenshot above was taken on JDK 6 Update 12, using the code from the article referenced at the start. I.e., without doing anything special at all, no fiddling with z-order or the like, the behavior was exactly as an end user would expect it to be. Hurray!