The "double Yamoto" is the monstership driven by a Mod. It is pretty much capable of defeating 20 BBs 6CVs and 10 SS plus all the CA, CL, and CV in a room all by itself. It carries Nukes. It has incredible repair speed and almost infinite ability to take hits.
It is in a "special" where if you are the one to sink the monster ship, you get a big reward.
The rubber duck is a Mod driven ship. In some cases, the special rulles may be "No Torps" so the rubber duck will sink any ship that fires torps. Otherwise it just monitors the game.
It is fun to fight the monster ship and I've played against it a number of times. I think I have seen it sunk once.
I don't believe I've seen it for about a year. Would be nice to have a monster ship battle again. Its a kick.