@jbuddy_13 I can only speak for myself: When a model has a lot of parameters and the posterior of each parameter can be expected to be approximately normal, then ADVI is much quicker than sampling. If needed, the result can be refined through sampling by drawing start values from the ADVI posterior. I find this to be convenient in many applications. However, you may find better answers in a separate discussion.
Furthermore, Bayesian NNs are already part of PyMC3: Variational Inference: Bayesian Neural Networks — PyMC3 3.10.0 documentation