Question:

Why is the tank shell and anchorage calculations based on empty weight of the tank while the water filled tank weighs significantly more?

Answer: 

For a Storage Tank (per design code, ex. API 650), the water height only induces a hydrostatic pressure on the wall, the weight of the water is a vertical force directed downwards at the base of the tank. 

Therefore, the weight of the empty tank is directed at the tank shell and anchorage locations around the perimeter of the tank.