Why can't airline pilots listen to music while flying?
I don't know where he gets that.  Regulation wise there's nothing stopping an airline pilot from listening to music as long as he/she can hear ambient sounds or communications and modern headsets make it easy to link an ipod to headphones that are also receiving the comms.

Pilots are allowed to take naps in flight as long as they are awake within 45 min of arrival.  A pilot not flying (PNF), where the autopilot is on and the other pilot is monitoring everything, can be doing paperwork or reading a paper or watching a movie on a tablet or going for a dump or wandering around the cabin if he/she wants.  Once on an arrival however, forget it and certainly not once below 10000 ft.

If there is a prohibition in that Capt's case, it is probably a policy with that airline.