This is actually a difficult engineering question. The superiority of a hyperboloidal shell over conical or cylindrical isn't clear cut, especially over a conical shell. But the chief difference is that at all surface points of both the cone and the cylinder have at least one principal curvature of zero, meaning that both can be "rolled out flat", or formed from a sheet of paper. The hyperboloid cannot be, so it resists local deformations everywhere on its surface, and hence it's "stiffer". Technically speaking, a hyperboloidal has a negative Gaussian curvature everywhere, while the other two have 0 Gaussian curvature everywhere.
A real hyperboloidal cooling tower is actually a compromise, because air accelerates as it rises inside the tower, and the cross section has to be reduced with height for efficient non-turbulent flow. Yet, a slight flaring at the top is found to be more efficient for dispersal of warm air. So, hyperboloidal cooling towers are generally wider at the bottom than at the top, and bottom part closely approximates a conical surface.
What the public isn't generally aware of is that hyperboloidal cooling towers are mostly empty space. All the cooling coils and surfaces are located at the bottom of the tower. The tower itself is designed to maximize air draft that would naturally occur with air rising after contact with heated surfaces.