The JRE is just the Java (OpenJDK) runtime to run Java apps, desktop apps, and server apps, but not applets. For applets you need to web browser plugin. The web browser plugin is provided by the package
I know where your confusion comes from: in the Windows world, "installing Java" gets you the JRE + the browser plug-in, which is essentially a part of the JRE. Well on Linux it's not, because when Sun open sourced Java under a GPL license back in 2006 creating the OpenJDK project, they did not include the browser plugin component.
I have argued with the JRE developers that Icedtea-web is counter-intuitive and that at least an alias of "openjdk-plugin" should be used, but met with staunch opposition from the Icedtea devs.... so there... Install icedtea-web.