We conduct a theoretical and numerical study of the aliased spectral densities and inverse operators of Matérn covariance functions on regular grids. We apply our results to provide clarity on the properties of a popular approximation based on stochastic partial differential equations; we find that it can approximate the aliased spectral density and the covariance operator well as the grid spacing goes to zero, but it does not provide increasingly accurate approximations to the inverse operator as the grid spacing goes to zero. If a sparse approximation to the inverse is desired, we suggest instead to select a KL-divergence-minimizing sparse approximation and demonstrate in simulations that these sparse approximations deliver accurate Matérn parameter estimates, while the SPDE approximation over-estimates spatial dependence.