A direct numerical simulation of the spherical Couette flow between two spheres with the inner sphere rotating was performed to investigate the detailed structure, formation process and mechanism of the spiral Taylor-Görtler (TG) vortices. For comparison with our previous experiments, a moderate gap case with clearance ratio β = 0.14 is chosen in the present numerical study. With adequate initial and boundary conditions, we have sucessfully simulated the supercritical spiral TG vortex flow in this system. Analysis of the numerical results reveals the structure and features of the spiral TG vortices. The flow consists of one toroidal TG vortex, one toroidal vortex cell, three spiral TG vortices and a secondary flow circulation in each hemisphere, and this supercritical flow solution features rotational and equatorial asymmetries. It is found that the spiral TG vortices are composed of a pair of counter-rotating, unequal spiral vortices with essentially different structural forms. One begins in the secondary flow circulation at higher latitude and ends with a connection to the toroidal vortex cell at lower latitude while the other one starts on the inner rotating spherical surface at lower latitude and ends on the outer stationary spherical surface at higher latitude. Through sucessive visualizations which display the transient features of the spiral TG vortices, we observe that vortex tearing, splitting, tilting, reconnecting, stretching and compressing occur in the formation of the spiral TG vortices. Pairing of two alternating helical vortices is the key process in their evolution. To understand the formation mechanism, we consider the vorticity production in the azimuthal vorticity component equation. The important vorticity tilting and stretching terms play different roles in the formation process of these two counter-rotating spiral vortices. The vorticity tilting term is responsible for generating both of the spiral vortices. The vorticity stretching term acts to stretch one of the spiral vortices from the inner sphere to the outer sphere while suppressing the stretching of the other in the azimuthal direction. The different formation mechanisms for these two counter-rotating spiral vortices lead to the structure of the spiral TG vortices.