The following thoughts are general, not necessarily entitled to every models.
The good (not exclusive)
1. probabilistic modeling, principled.
The good (exclusive)
1. generative process enables to plug in “domain knowledge” very easily.
2. prior enables to plug in further “domain knowledge”.
3. integration/summation over latent variables usually yields better performance.
The bad (not exclusive)
1. setting prior can be difficult and may require hyper-parameter turning.
The bad (exclusive)
1. the specific generative process with “domain knowledge” can lower the model’s ability to explore complicity in the data.
2. the design of generative process requires deep understanding about both data, “domain knowledge” and math.
2. the inference is complicated and slow, especially with integration/summation over latent variables.