We used to do it because dealing with cross-browser compatibility was a pain. For instance, I used to use a 1-pixel applet instead of XMLHttpRequest for AJAX (slightly before they coined the term "AJAX") and applets for rendering live-updated XML-based diagrams (instead of SVG) in both cases because it simply made things less buggy and less of a pain to support at the time. But that was in 2004 to 2007.

But once we've written something, and it works, we're often a little slow updating it to newer tech, because we can't write a paper out of "the same thing I published earlier, but now using SVG and XHR instead of applets".

If it still just-about works, there's less impetus to rewrite it.

There are quite a few applets demonstrating chemical reactions/bonds or physical equations etc., and as such applets are relatively common in teaching materials for high school (/secondary education) physics and chemistry, at least with teachers (I had or) I have substituted.

They are obviously years old and rarely signed, which is a shame because quite often there are plenty of demos, none of which I can run with modern default settings of a PC (and sometimes I wouldn't have enough credentials to do anything about it, anyway).