Pretty sure nothing comes of that after the hunt is over. All that really goes on if Zidane wins is he gets 5000 gil, and if Vivi wins he gets a card. Can't remember which one it is though.
After the winner of the Hunt receives their reward, that wounded Burmecian shows up and tells everyone that the Kingdom has fallen under attack by a bunch of Black Mages.
That's what I remember of it anyway. Can't seem to recall any other time in the game where the outcome of the Hunt was of any importance.