The fundamental difference between an app and an applet is that an app is supposed to run on a standalone computer whereas an applet requires a browser for it to function.
Basically, an applet is a web-version of an app, so to say.
An app always begins with the main method whereas there is no main method for an applet. The browser requires a java-plugin for it to run an applet.
There is another category called servlets, which are just applets that run on the server side, whereas the normal applets run on the client-machine. Both applets and servlets work together to accomplish the given task!
Almost all the flash games that you play online, or the various online forms, are simple applets.
Examples of apps include every software that you can run on your computer without having to use a web browser.