Browsers forum: Using Java with Firefox & Internet Explorer

I am using Windows XP and Firefox 3.8.6, the newest version.

I wanted to use a learning tool from a website. However, the message at that website said This page contains a Java applet that your browser is unable to display. Please go to http://java.sun.com/getjava/ to download free plug-in
At that next page, the download is for Version 6 Update 21.
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With all the known issues in using Java, especially with Firefox, would updating my Java applet cause problems? Or should I try to update my MS Internet Explorer browser? I would have already done this if I were sure updating MS-IE would not also affect Firefox.

Both those browsers work fine right now. Id hate to fix something that isnt broken but I would like to see and use that online training tool. Thanks.
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Note: For convenience, below are some issues from the Mozilla website of using Java with Firefox. Do any of these appear to affect my wanting to update my current Java applet in Firefox?

Java Known Issues
http://plugindoc.mozdev.org/faqs/java.html#Issues
General
Support for the classic Java plugin has been dropped in Firefox 3.1a1 and later [Bug 435334]. Workaround: Use the Java Runtime Environment 6.0 Update 10.
Firefox 3.0: If Java is uninstalled, the plugin finder service will not prompt to reinstall it [Bug 433502]
Microsoft Windows
Remove any Java plugin files from your Plugins directory when installing a new JRE release (if present).
To speed the Java plugin up on Windows, use the Java Control Panel to disable the Java icon appearing in the system tray, and disable the Java console.
Using Adblock Plus or NoScript with Firefox 3.0 may cause crashes when loading Java applets [Bug 409566]
The Java plugin may not work if HKLM\Software\mozilla.org\Mozilla\CurrentVersion is missing or inaccessible [Bug 400878, Sun Bug 6620045]
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