The main difference between a variational auto-encoder (VAE) and an auto-encoder is that the VAE is a generative and statistical model while an auto-encoder (AE) is just a data compressor and decompressor (it is just a function approximator), so an AE is not a statistical model. They both have an encoder and a decoder and they both convert the inputs to a latent representation, but their inner workings are different and their purpose is also slightly different. An AE could be considered a generative model, in the sense that it generates objects that are similar to the input (but they might not be the same as the input). However, it is not a generative model according to the usual statistical definition of a generative model.
In the case of an AE, given an input $x$ (e.g. an image), the encoder produces one latent vector $z_x$, which can be decoded into $x'$ (another image which should be similar or related to $x$). Compactly, this can be presented as $x'=f(z_x=g(x))$, where $g$ is the encoder and $f$ is the decoder. This operation is deterministic, in the sense that, given the same $x$, the same $z_x$ and $x'$ are produced.
In the case of a VAE, given an input $x \in \mathcal{X}$ (e.g. an image), more than one latent vector, $z_{x}^i \in \mathcal{Z}$, can be produced, because the encoder part of the VAE models the inputs as a probability distribution. In other words, the encoder of a VAE assumes that the inputs $\mathcal{X}$ are drawn from a probability distribution, which can be denoted by $\mathcal{X} \sim q$, where $q$ is some computable probability distribution (e.g. a Gaussian distribution with mean zero, $\mu =0$, and a finite standard deviation, $\sigma$). $q$ is defined over $\mathcal{Z}$ (the space of possible latent vectors). More specifically, $q$ is a conditional probability distribution, which can be denoted by $q(z \mid x)$.
So, the encoder of a VAE, while being trained, creates a probability distribution, which, in practice, can be represented by neural networks, that is, you can train a neural network to represent the parameters of e.g. a Gaussian distribution (which is defined by two parameters: the mean and standard deviation). You can thus more precisely denote $q$ by $q_\phi$, where $\phi$ are the parameters of a neural network that e.g. output two numbers: the mean and a standard deviation of $q$.
Given a probability distribution $q_\phi$, we can sample several $z_{x}^i$ from $q_\phi$. Why is this useful? Given the same input $x$ (e.g. a graphical representation of a molecule $m$), we can sample e.g. $N$ latent vectors $z_{x}^1, z_{x}^2, \dots, z_{x}^N$, which can all be decoded (e.g. into $K$ new molecules). Why would this be useful? For example, in the case you need to generate molecules which have a similar structure to the input molecule $m$. Why would you need this? In some cases, molecules with a similar structure can have similar effects, but could e.g. be more easy or inexpensive to synthesise.
Why can we generate $K$ new molecules out of $N$ latent vectors, where $K$ can be greater than $N$? We have not yet mentioned the details of a decoder of a VAE. The decoder of a VAE also models its inputs, $\mathcal{Z}$, as a probability distribution defined over $\mathcal{X}$, which can be denoted by $p_\theta(x \mid z)$, where $z$ is a latent vector. In other words, given the $i$th latent vector sampled from the latent distribution (the distribution of the encoder), $z_{x}^i$, we can sample more than one reconstructed input from the probability distribution of the decoder. Hence, a VAE often constructs two probability distributions: one that is associated with the encoder, $q_\phi(z \mid x)$, and one that is associated with the decoder, $p_\theta(x \mid z)$.
Given that VAEs create these distributions, they are considered statistical models. An auto-encoder has no notion of a probability distribution. An AE only maps inputs to outputs deterministically, so it is a function approximator (given that it approximates a function that compresses the input and then another function that decompresses it). VAEs are thus a lot more flexible than AEs.
