What is the difference between causal models and directed graphical models?

Causality by Judea Pearl is the book to read.

The difference is that one is causal and the other is merely statistical. Before dismissing me as a member of the tautology club, hear me through.

A directed probabilistic relationship (AKA a complete set of Conditional Probability Tables , AKA Bayesian Network) only contains statistical information. Meaning that anything you can infer from the Joint Probability table you can infer from the directed probabilistic relationship, nothing more, nothing less. The two are equivalent.

A causal relationship is something else entirely. A causal relationship (AKA Causal Bayesian Network) must specify what happens under any variable intervention. Intervention is when a variable is forced to a value outside of the normal influences of the model. This is equivalent to replacing the conditional probability for the forced variable (or variables, but we consider just one for simplicity) with a new table in which the variable takes its forced value with probability one.

If this does not make sense, please follow up and I will clarify.

This section added to address Neil's questions in the comments

Neil asks:

How can you determine the direction of directed probabilistic relationships without performing interventions? In other words, doesn't the directed graphical model have causal information in it (i.e., information about probabilities conditional on interventions?)

You can determine the direction of directed probabilistic relationships by making additional non-statistical assumptions. These assumptions commonly include: assuming no hidden variables, and the really important one, assuming that the conditional independence relationships found in the joint distribution are stable (meaning they exist not by chance or cancellation). Bayesian Networks do not make these assumptions.

For details of how to recover the directions research the IC, PC, and IC* algorithms. I believe the specific details of IC are covered in: "A Theory of Inferred Causation"

There are two types of causal model: interventional models and counterfactual models. All directed graphical models can reason observationally. An interventional model is a directed graphical model that can reason with observational and interventional evidence. A counterfactual model can reason with observational, interventional, and counterfactual evidence (interventions whose source is inferences within the model).

In a private email a couple years ago, Pearl wrote me that:

By definition, a model is a list of assumptions, and assumptions are never "known to be true". They may be substantiated by theory, or data, or experiments. But their position in the hierarchy is determined by what they claim, not by where they came from.