Is there a formal difference between the terms "Probabilistic Model" and "Statistical Model"? Is there a methodological difference between the two or just a preference of terminology? I see the former used most often with graphical models and Bayesian models, while "statistical" seems to be anything specifically Frequentist.
Stats textbook are sometimes divided into two sections: 1) Probability and 2) Statistics.
Probability was about permutation, combination, conditional probability. Probability was often explained with dice, coins, colored marbles and other discrete artifacts. Probability is the measure of the likelihood that an event will occur. Although probability can be calculated using statistical models, and probability does not have to be from countable events and a rational number on 0-1, that would be its first meaning. Statistics was about mostly real number measurements, like the t-, z-, W- and U- statistics. A statistic (singular) or sample statistic is a single measure of some attribute of a sample (e.g., its arithmetic mean value).
There is an overlap or gray zone for probability for the discrete distributions including the binomial distribution, the multinomial distribution, and the Poisson distribution, which are still finger counting, i.e., literally countable probable events, for which statistics are parameters and for statistics for continuous distributions that are real number models of probability.
Probability models as a first meaning imply countability, for example, the likelihood of getting exactly 5 heads from 10 coin tosses. Statistical models, as a first meaning contain some statistic. However, one can say that "Marginal probability is a statistic" so that using which phrase is used when depends on what connotation one is making.