I think the big advantage is that Flash is installed on a huge percentage of computers, whereas any given user may have Windows Media Player, but not Quicktime, or Quicktime but not RealPlayer, etc. It's a more consistent way of presenting movies, so users aren't forced to select a player beforehand. It's also more convenient for developers, since we only have to embed one type of object.
If you have Flash 8, they're not too complicated at all. It comes with lots of nice Actionscript functions for handling the movies, and the support available seems pretty good. On the other hand, you can just go download a player that will take the URL of your FLV movie and play it - someone mentioned flvPlayer in another thread.