Yes, the two are orthogonal concepts. Text did not disappear just because we invented photography. Bayesian data analysis is for inverse problems, such as using data to learn about the properties of the system/model that could have generated the data, and neural networks are for forward problems such as using data to generate more data or make predictions.
You can use BDA for forward problems too, via posterior predictive samples. The benefit over neural networks for this task is that with BDA you get dependable uncertainty quantification about your predictions. The disadvantage is that the modalities are somewhat limited to simple structured data.
You can also use neural networks for inverse problems, such as for example with Neural Posterior Estimation. This approach shows promise since it can tackle more complex problems than the standard BDA approach of Markov Chain Monte Carlo and with much faster results, but the accuracy and dependability are still quite lacking.
You can use BDA for forward problems too, via posterior predictive samples. The benefit over neural networks for this task is that with BDA you get dependable uncertainty quantification about your predictions. The disadvantage is that the modalities are somewhat limited to simple structured data.
You can also use neural networks for inverse problems, such as for example with Neural Posterior Estimation. This approach shows promise since it can tackle more complex problems than the standard BDA approach of Markov Chain Monte Carlo and with much faster results, but the accuracy and dependability are still quite lacking.