We have studied the kinetics of domain growth in the (4×1) uniaxial [or (2,2)] phase of the two-dimensional anisotropic next-nearest-neighbor Ising (ANNNI) model with Monte Carlo methods using Glauber dynamics. The growth is shown to be spatially anisotropic, with the anisotropy depending strongly on the anisotropy parameter α. In addition to this, a more abrupt change is found as one crosses a wetting transition line in the model. Despite this a dynamical exponent n≃0.5 is obtained at low temperatures for all values of α. To explain these results, a phenomenological theory of domain growth developed originally for the clock model is extended to include the uniaxial (4×1) phase. In particular, it is demonstrated that the more abrupt change near the wetting transition occurs due to the disappearance of a vertex-antivertex configuration present in the dry region. Also, the ANNNI model with conserved dynamics is shown to belong to a different universality class than a model with a symmetric p=4 phase studied recently.