First, they not gone. You can still find lot of applets on the Web, lot of people use them particularly to demonstrate algorithms and such.
Advantages: can leverage existing libraries (math, physics, sorting, graph, etc.) and it is faster than Flash.
Inconvenience: it might be risky to target a recent JVM (although Sun did a good job on automatic updates, looks like lot of people are using Java 1.6 already), load time is a bit slow (even though great progress have been made there).
You can still find lot of game applets too, like Bookworm, with the added advantage, perhaps, to have part of work already done to run them on mobile phones...
Second, I can predict a regain of interest with JavaFX. Applets on steroids, able to break legend of "applets are ugly"... :-)
Last, a library like Processing makes super easy to create graphical intensive applets, and you can find lot of them on the Net, eg. on OpenProcessing where the worst (beginners in programming) is near the best!