Computational fluid dynamics (CFD) of industrial-scale, axial compressor geometries has traditionally been performed using steady state methods such as the mixing plane approach. With the surge in the development of large-scale, massively-parallel computing platforms, fully 3D unsteady approaches are rapidly growing in popularity. The fully 3D, unsteady approach involves building a full 3D domain for each blade row, and then coupling the stationary and rotating domains using a sliding interface. In the literature, there are various methods for solving this 3D unsteady problem, such as the Unsteady Reynolds Averaged Navier-Stokes (URANS) and the Detached Eddy Simulation (DES) methods. While these methods are well documented for a variety of real-world problems, there have been limited efforts to compare the effectiveness of these methods for fully 3D, unsteady turbomachinery problems. In this study, the first stage of an industrial-scale axial compressor was simulated using: i) the URANS approach, and ii) the DES approach. The compressor geometry consisted of an inlet housing, inlet guide vanes (IGV), a rotor, and a stator. The RANS model for both simulations was the k-epsilon model. For both of these cases, sliding mesh interfaces were located between the IGV and rotor, and between the rotor and stator. The results of the URANS and DES approaches were time-averaged and their predictions were compared. Throughout the study, our goal was to provide important insights into the performance of the URANS and DES approaches, and to highlight the essential differences.