By using the phase-field approach, we have simulated the directional growth of alloys in undercooled molten states under the isothermal and nonisothermal conditions. The influences of the isothermal approximation on simulation results are discussed. We found that for undercooling greater than 25K, the isothermal approximation overestimates the interface growth velocity and reduces a critical velocity for an absolute stable planar interface, thus in this simulation,the interface morphology shows the plane-cell-plane transition with increasing initial undercooling of the melt, and the planar interface obtained under a large undercooling is absolutely stable. Whereas in the nonisothermal simulation,only plane-cell transition occurs in the same range of the initial undercoolings of the melt, and the planar interface tends to be destabilized and evolve into cells.