The properties of a dual-frequency vertical external cavity surface emitting laser (VECSEL), in which two linear orthogonal polarization modes are oscillating simultaneously, are theoretically investigated. We derive a model based on the ideas introduced by San Miguel et al. [San Miguel, Feng, and Moloney, Phys. Rev. A 52, 1728 (1995)], taking into account the spin dynamics of the carriers inside the quantum well (QW) based gain medium of the dual-frequency VECSEL. This model is shown to succeed in describing quite a few properties of the dual-frequency VECSEL, such as the behavior of nonlinear coupling strength between the modes, the spectral behavior of intensity noises of the modes, and also the correlation between these intensity noises. A good agreement is found with experimental data. The variables associated with the spin dependent carrier dynamics can be adiabatically eliminated due to the class-A dynamical behavior of the considered laser which is based on a cm-long external cavity. This leads to a simple analytical description of the dynamics of the dual-frequency VECSEL providing a better understanding of the physics involved.