As others have said, less handling of the beer by the brewer. That means less chance of infection and less chance of aeration. No siphoning is a BIG advantage. Just put your keg under the conical, open the valve and let it go (attach a hose too of course).
For harvesting yeast, I've read in a few different places that a 60 degree cone is optimal. To harvest yeast, simply sanitise the bottom port, put a sterilised vessel under it, and open the valve a bit. To remove the yeast from the vessel, do the same thing until beer starts to come out. You have to close & open the valve quite a few times to allow the yeast to fall down into the bottom, but it works great.
Besides, conicals are what almost every commercial brewery use, so they must have an advantage.