A Distinction between Causal Effects in Structural and Rubin Causal Models
Structural Causal Models define causal effects in terms of a single Data Generating Process (DGP), and the Rubin Causal Model defines causal effects in terms of a model that can represent counterfactuals from many DGPs. Under these different definitions, notationally similar causal effects make distinct claims about the results of interventions to the system under investigation: Structural equations imply conditional independencies in the data that potential outcomes do not. One implication is that the DAG of a Rubin Causal Model is different from the DAG of a Structural Causal Model. Another is that Pearl's do-calculus does not apply to potential outcomes and the Rubin Causal Model.
Aliprantis, Dionissi. 2015. “A Distinction between Causal Effects in Structural and Rubin Causal Models.” Federal Reserve Bank of Cleveland, Working Paper No. 15-05. https://doi.org/10.26509/frbc-wp-201505