The synthetic fiber rope is a mooring material that is increasingly used, but few commercial codes are known to reliably implement the nonlinear stiffnesses of this material. Through the development of a floating platform-mooring line coupled simulation code that allows simultaneous consideration of the strain- and strain rate-dependent nonlinear stiffness, this paper aims to examine the effects of the nonlinear stiffnesses on platform motion and mooring tension responses. We introduced theoretical background to develop a new code. We compared the experimental and numerical results of two floating platforms with the results calculated from the code. A very good agreement was drawn from the comparative study for the two verification platforms. We analyzed the effect of the nonlinear mooring line stiffness on the platform motion and mooring line tension responses by assuming three mooring materials with the linear, strain-dependent nonlinear, and strain rate-dependent nonlinear stiffnesses. When applying the catenary mooring system, the nonlinear stiffness little affects the motion and line tension responses. Considering the nonlinear stiffness in the taut-leg mooring system, the slowly varying responses in the surge motion and mooring tension were observed in the response spectra. The strain rate-dependent nonlinear stiffness should be considered for the taut-leg mooring system.