I think you have it backwards. And it is not so much a difference between HSPICE and Spectre. Rather many people model nonlinear capacitors in the same way they model linear capacitors, but in doing so they violate the basic assumption that capacitance does not change that was used when creating the linear model in the first place. This is explained in Modeling Varactors.

A nonlinear capacitor model that is specified in terms of its capacitance is always problematic. To get good results you need to analytically integrate the nonlinear capacitance with respect to voltage to get the charge of the capacitor as a function of voltage. Then you calculate the current through the capacitor by taking the derivative of the charge with respect to time.

-Ken