What is a statistical model?
According to Anthony C. Davison (in the book Statistical Models), a statistical model is a probability distribution constructed to enable inferences to be drawn or decisions made from data. The probability distribution represents the variability of the data.
Are all neural networks statistical models?
All neural networks that construct a probability distribution to draw inferences from the data or to make decisions from the data are statistical models.
Variational auto-encoders (VAEs) construct a probability distribution (e.g. a Gaussian) to draw inferences, so VAEs can be considered statistical models.
On the other hand, for example, MLPs do not necessarily construct any probability distribution, so they are not necessarily statistical models. However, note that MLPs can be used to represent the parameters of a distribution. For example, you could train a MLP to represent the mean of a Gaussian distribution. See e.g. Junction Tree Variational Autoencoder for Molecular Graph Generation for an example.
Consequently, not all neural networks are statistical models (at least, according to the definition by Davison).