  Most of the scientific literature on causal modeling considers the structural
framework of Pearl and the potential-outcome framework of Rubin to be formally
equivalent, and therefore interchangeably uses the do-notation and the
potential-outcome subscript notation to write counterfactual outcomes. In this
paper, we agnostically superimpose the two causal models to specify under which
mathematical conditions structural counterfactual outcomes and potential
outcomes need to, do not need to, can, or cannot be equal (almost surely or
law). Our comparison reminds that a structural causal model and a Rubin causal
model compatible with the same observations do not have to coincide, and
highlights real-world problems where they even cannot correspond. Then, we
examine common claims and practices from the causal-inference literature in the
light of these results. In doing so, we aim at clarifying the relationship
between the two causal frameworks, and the interpretation of their respective
counterfactuals.
