I think, I finally found an answer to the question, why one should use Flash instead of HTML5: Binary sockets
Newer Flash Player versions allow developers to open binary socket connections, which is not possible using pure HTML5/JavaScript in the browser (HTML5 only supports websockets, which are different).
Thus, for example, it would not be possible to create a FTP client without Flash or a backend system since such an attempt would require one to open a socket in order to implement the FTP protocol.
