Numerical analysis of the spherical Couette flow of a viscoelastic fluid during rotation of the inner sphere is performed using a constitutive equation from either the Giesekus model or the Oldroyd-B model. The numerical solutions for the flow field are obtained using the finite-difference method with a decoupling technique. The transient torque characteristics associated with the flow field that results from rotation of the inner sphere are also calculated. Two basic flow behaviors, which are obtained from either the Giesekus model or the Oldroyd-B model, are compared with Newtonian flow behavior. The numerical simulation reveals that the shear-thinning effect on the shear viscosity strongly influences the flow characteristics in the equatorial region in the Giesekus model. At a critical Reynolds number, the flow becomes unstable, forming elongated Taylor-Gortler vortices of different sizes. With the Oldroyd-B model it was shown that at a high Deborah number, elastic instability gives rise to a substantial change in the flow mode in the transient process of the flow behavior.