Log-likelihood could be wrong so comparing covariance pattern models based on log-likelihood and LR tests isn't a perfect way.
Intuitively, how do I know when to use Toeplitz versus AR(1) versus Banded Main Diagonal, Toeplitz with 2 bands, , spatial power, ARMA, First order first factor analytic, Huynh-Feldt, First-Order Ante-dependence, Direct Product AR(1)?
Of Course AR(1) is like markov-chain decaying correlation. Unstructured is no assumptions. Compound symmetry is same correlation at every time point.
SAS has a huge list of covariance pattern structures I referenced.