The main thing is that these plugins how Java provides almost unrestricted access to your computer. This is great "advantage" of having an application running Java in the browser. At least that was the reason in the past.
The browser is extremely limited when it comes to doing anything outside of it, outside draw a page and make basic communication within certain limits with the server where it is chatting 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 browser "pure".
In addition to this usually the language and libraries available are usually a little more powerful giving more tools for the programmer to work. 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 gender but the choice is not usually made because of this.
There is a tendency to no longer use this type of feature because it even helps to bring security problems to the machine of its users. Also many people can’t even get along with the installation of plugin and either give up or pay a technician to do for them.
If you consider that sites that need this today have been replaced by native mobile apps they should probably have native applications for major operating systems desktop and eliminate this kind of need. So would be in the website only the basic functionality that the 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 less problem you have to install an application that the plugin.
One reason they still use Java is because in the past they used Java. So they don’t want to do anything new. There was a reason to use Java but no more.
In addition many programmers choose to do it in Java because that’s what they know or like. Even if you don’t need it specifically, you don’t need to "rotate anywhere," it’s chosen. This, of course, can apply to any language or platform.
Some people find the "Compile philosophy once on every platform and run anywhere" found in C/C++ and other languages to be better and more portable. Of course there may be disadvantages too.
Note that there are restrictions on these plugins, they fell into disuse and since version 9 of Java it is considered deprecated and should no longer be used, and Java itself is heading for another model better than the original even when it was not the plugin.
Relating: do the commonly used security modules on bank websites?
– rray