  We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL)
vector model for nematic liquid crystals, confined in a two-dimensional square
domain with tangent boundary conditions. An efficient numerical algorithm is
developed to construct the solution landscapes by utilizing the symmetry
properties of the model and the domain. Since the sEL model and the reduced
Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals,
we systematically compute the solution landscapes of the sEL model, for
different domain sizes, and compare with the solution landscapes of the
corresponding rLdG models. There are many similarities, including the stable
diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes
with low-index saddle solutions. Significant disparities also exist between the
two models. The sEL vector model exhibits the stable solution $C\pm$ with
interior defects, high-index "fake defects" solutions, novel tessellating
solutions, and certain types of distinctive dynamical pathways. The solution
landscape approach provides a comprehensive and efficient way for model
comparison and is applicable to a wide range of mathematical models in physics.
