Based on the lasing threshold equation of a diode-end-pumped Nd:YAG laser, the threshold condition of simultaneous dual-wavelength lasing is determined and the relationship between the transmission ratios of the output mirror corresponding to the two wavelengths is calculated. A simultaneous dual-wavelength continuous wave (cw) or quasi-cw diode-end-pumped Nd:YAG laser operating at 1319nm and 1338nm has been demonstrated. A total cw output power of 6W was achieved at a pump power of 21.6W with a slope efficiency of 30%. And a total quasi-cw output power of 4.75W was obtained at a pump power of 21.6W and a pulse repetition rate of 50kHz, with a slope efficiency of 24.73%, and the corresponding pulse width was 55.05ns. Inserting a Brewster plate in the cavity and the highest linearly polarized wave output power was boosted up to 2.22W at a pulse repetition rate of 50kHz. The instability of the output power was less than 0.52% and the beam quality M2 factor was as low as 1.16 at the linearly polarized wave output power of 2.17W. The simultaneous dual-wavelength laser provides an experimental basis for generating highly coherent terahertz wave radiation of 3.23THz by nonlinear optical difference frequency method.