Is it permissible for the older brother to destroy the music tapes of his younger brother who neglects prayer and listens to a lot of music, knowing that the younger brother does not respond to advice and gets angry at such an action?

You should advise your brother, preach to him, and invite him to Allah with wisdom and gentleness. Explain the immense status of prayer and the danger of abandoning it, as well as the peril of listening to music, which deadens the heart and weakens faith. You should also give him beneficial tapes and explain that true happiness is achieved by adhering to Allah's Sharia. It is not permissible to help him obtain forbidden things, nor to pressure him to do so. Destroying music tapes is permissible unless it leads to a greater harm, but it is preferable to resort to it at a later stage.