A causal model goes beyond the graph by including specific probability functions for how to calculate the probability of each node taking on the value given the values of 's immediate ancestors\. It is implicitly assumed that the causal model factorizes, so that the probability of any value assignment to the whole graph can be calculated using the product:
No, this kind of factorization is used for any probabilistic graphical model (PGM), whether or not it is causal. The difference is that for a causal model an arc from node x to node y additionally indicates that x has a causal influence on y, whereas there is no such assumption in general for PGMs.