The reason for using a dynamical grid rather than a square grid is the same as why people working in fluid dynamics use adaptive grids rather than square ones - they adapt much better to the solution. Even if one starts out with a regular grid, different regions very soon need different resolution, which can be captured numerically (without unduly large work) only with an adaptive grid.
To get the same numerical accuracy with a square grid one would need an much finer (indeed typically extremely fine) lattice spacing.
A second reason is that in general relativity one wants to consider topologies different from the topology of Minkowski space - and possible even changes in topology. This cannot be done by a square lattice. On the other hand, a triangulation of a manifold completely describes its topology, hence is the appropriate tool for the discretization of general manifolds. A manifold in itself has not yet metric propoerties such as curvature, but these are given by fields defined on lower-dimensional subsimplices. This generalizes lattice gauge theory where the gauge field is given on the edges of the lattice, rather than on its vertices.