A smooth and discontinuous (SD) oscillator is a typical multi-stable state system with strong nonlinear properties and has been widely used in many fields. The nonlinear dynamic characteristics of the system have not been thoroughly investigated because the nonlinear restoring force cannot be integrated. In this paper, the nonlinear restoring force is represented by a piecewise nonlinear function. The equivalent coefficients of fractional damping are obtained with an orthogonal function. The influence of fractional damping on the transition set, the amplitude–frequency response and the snap-through of the SD oscillator are analyzed. The conclusions are as follows: The nonlinear piecewise function accurately mimics the nonlinear restoring force and maintains a nonlinearity property. Fractional damping can significantly affect the stiffness and damping property simultaneously. The equivalent coefficients of the fractional damping are variable with regard to the fractional-order power of the excitation frequency. A hysteresis point, a bifurcation point, a frequency island, pitchfork bifurcations and transcritical bifurcations were discovered in the small-amplitude resonant region. In the non-resonant region, the increase in the fractional parameters leads to the probability of snap-through declining by increasing the symmetry of the attraction domain or reducing the number of stable states.