> A statistical model is a mathematical model which is modified or trained by the input of data points.

> A probabilistic model specifies a probability distribution over possible values of random variables, e.g., P(x, y), rather than a strict deterministic relationship, e.g., y = f(x).

For example, by these definitions, we could use a linear regression as a statistical model that is not a probabilistic model; we could make a bayes network (using known distributions) as a probabilistic model that is not a statistical model; and we could make a Hidden Markov Model trained on sample data that would be both a statistical and probabilistic model.

In the linear regression model the points (x,y) have the following relationship (in the most simple example)

> y_t = x_t + e_t

where e_t is some error term. Would this be a statistical or probabilistic model?

We do know that the best predictor is just the conditional expectation (in a linear setting bla bla)

> y_pred = E[x_t + e_t | .. ] = x_t

Or is this what you mean with "model"? The predictor? Sorry for being a bit confused.

Usually, every statistical model is a special case of a probabilistic model in which its parameters have been estimated by data.

The regression E[Y|X] is basically the mean of a gaussian distribution of Y given X with sigma set to the error term. The whole gaussian distribution part is the probabilistic model. But to estimate wich parameters make up this model in a particular application (together with how to check if it is valid or not) is the statistical modeling part.