In my opinion Java Applets have been dead for years. I wrote some in the late 90s - a Tetris game during an internship to demonstrate on a 40MHz ARM Acorn Set Top Box for example. Of course I bet there are some casual game sites that have tonnes of them still, and thus it will remain supported, but active development will/has dropped off.
Java Web Start is a handy technology in my opinion. That will still work with JavaFX, it's just another library for that system.
JavaFX will give Java opportunities beyond technical tools (like SQL Developer), in-house business applications and server applications (where it excels). I think it's one of those libraries that is worth learning for any Java developer, if they can get the time. There's no arguing that user interface libraries for Java have been sorely lacking, or overly complex, for many a year.
However there's a lot of competition out there, and it is very new (which means the development tool support is very raw, compared to Flash and Silverlight). Also people don't like downloading massive runtime environments, although broadband makes it less painful than 5 years ago for many!