The main thing is that these plugins like Java provide almost unrestricted access to your computer. This is great "advantage" of having an application running Java in the browser. At least that was the motive in the past.
The browser is extremely limited when it comes to doing anything outside of it, outside drawing a page and making basic communication within certain limits with the server where it is talking in that context.
Of course it is common to have some limitations, these plugins still run on virtual machines but the limitation is much smaller than the "pure" browser.
In addition, the language and libraries available are usually a little more powerful, giving more tools for the programmer to work with. This has been more true in the past, but there is still more functionality in these plugins .
It is also possible to achieve better performance with Java or some plugin of the genre but the choice is usually not made because of this.
There is a tendency to no longer use this kind of feature because it even helps bring security issues to your users' machine. Also many people can not even understand the plugin installation and either give up or pay for a technician to do it.
If you consider that the sites that need it today have been replaced by native mobile apps, you should probably have native apps for the major desktop operating systems and eliminate this kind of need. This would give site only the basic functionality that web can provide by default. And an even more powerful and perhaps safer application can be used when you need something more powerful.
It will probably give you less trouble to have to install an application than the plugin .
One reason to still use Java is because in the past they used Java. So they do not want to do anything new. There was a reason to use Java but there is more.
Furthermore, many programmers choose to do it in Java because that's what they know or like. Even if you do not need it specifically, you do not need to "rotate anywhere," it is chosen. This, of course, can work for any language or platform.
Some people think that the philosophy of "compile once on every platform and run anywhere" found in C / C ++ and other languages is better and more portable. Of course there may be disadvantages too.