Java Applets are rarely used today except in giving 3D userface and some web applet chats.
While Flash plugins can be automatically downloaded and installed by the browser, the applet cannot.
The end users must already have installed JVM.
In my opinions, why AJAX can probably be over flash and java applet,it can extends its funtionalites as much as you like. No limit!
I mean,the page loads ajax js as much as it requires. Then reload it again when it's needed. If no need or no interaction from the users,it takes no actions.
The main benefit is if we use AJAX, we can make our site synchronous.
The good examples are Google Mail, Google Reader,Google Page Creator and Yahoo Live Search Beta.
Now you can implement Ajax in flash, which is called AFAX ::
AFLAX (Wikipedia Entry) stands for Asynchronous Flash and XML. Defined simply, AFLAX is a development methodology which combines Ajax and Flash to create more dynamic web based applications.
Developed by Paul Colton, the AFLAX technology is available as a library that enables developers to use JavaScript to fully utilize all of the features of Adobe's Flash runtime -- including graphics, networking, video and camera support.
http://www.aflax.org/
You can study some of good AJAX sites at ::
http://www.ajaxian.com/ http://www.ajaxblog.com/ http://www.fiftyfoureleven.com/resources/programming/xmlhttprequest/examples http://backbase.com Its like the normal win app on the web using AJAX.
http://zerobase.jp/blog/entry-275.html You can edit the their sites using AJAX
Google suggest is the latest real time search engine as you type.
http://www.google.com/webhp?complete=1&hl=en Here is the IME ( input method editor for japanese)
http://chasen.org/~taku/software/ajax/ime/ Here is Handwriting pad using AJAX
http://chasen.org/~taku/software/ajax/hwr/ Here is post code to address programm, its really work (by typing 7 characters )
http://ajasql.org/sample/0/sample.htm here is the other ajax site from eSkiSo.
www.ajaxwrite.com