which are the advantage of a variational autoencoder respect to a traditional neural network? I'm studying and implementing a variational auto encoders to perform unsupervised clustering. I understand that them use a probabilistic approach respect to a traditional neural network, but I don't understand which are their advantage respect to the traditional ones. Thanks for your answer

Variational autoencoders (VAEs) do ordinary variational inference (VI): You are given a joint distribution $p_\theta(x, z)=p_\theta(x|z)p_\theta(z)$, with unknown parameters $\theta$ and the goal is to find for the true posterior $p_\theta(z|x)$ an approximation $q_\phi(z|x)$, where $q$ has some easy, tractable structure. This is an old problem in machine learning, studied long before deep neural networks were known.
The only difference of the variational autoencoder approach to VI is, that the posterior $q_\phi(z|x)$ and the likelihood $p_\theta(x|z)$ are modeled as density neural networks, i.e. as deep neural networks that output the density parameters for the likelihood and the posterior. E.g., if you want to create a normal density, the output of the neural network would be the mean and the covariance matrix of the estimated normal. The network giving the parameters $\phi$ for the posterior is the encoder and the one providing the parameters $\theta$ for the likelihood is the decoder.
Now, since deep neural networks can be very powerful, one expects VAEs to often do better than traditional approaches to VI. Thus, whenever you have a VI problem, you might want to give VAEs a try.
However, you stated your problem as unsupervised clustering. But you did not state how you want to use VI for clustering. Maybe you want to cluster the latent variables $z$ and you hope that this provides some clustering information for the observed variables $x$, as investigated in this paper. Also, it is not clear what "traditional" neural network unsupervised clustering methods you would like to compare the VAE approach to.
Another problem that makes it difficult to answer your question is that it is hard to evaluate clustering methods. There are some clustering metrics but those are rather difficult to interpret and often inconclusive.
But, as a rather general and not very precise statement, I can say that I have not had much success with applying VI to clustering problems, with or without VAEs, and I also don't have the impression that this method is considered particularly competitive in the community. Yet, as always, the results very much depend on your scenario, and thus your mileage may vary.
