What do you mean by "Hierarchical Bayesian spatiotemporal model"? If I would fit a "Bayesian" model the data with a poisson model, "random effect" and an autoregressive component, I could call it a "Hierarchical Bayesian spatiotemporal model".

If use a a Generalized-Linear-Mixed-Model (GLMM) with poisson error distribution and autoregressive component I would fit the same model. However, I would only estimate the maximum likelihood, not the posterior distribution. Some more information (reference to Hierarchical Bayesian spatiotemporal model) would be needed to properly answer your question in detail.

Luckly both the "Bayesian" and "Frequentist" fit GLMMs. The difference is the inclusion of prior information and interpretation of the results. Unfortionately technical terminology make it for many difficult to distinguish.

Best,