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We examine counterfactual explanations for explaining the decisions made by model-based AI systems. The counterfactual approach we consider defines an explanation as a set of the system's data inputs that causally drives the decision (i.e., changing the inputs in the set changes the decision) and is irreducible (i.e., changing any subset of the inputs does not change the decision). We (1) demonstrate how this framework may be used to provide explanations for decisions made by general, data-driven AI systems that may incorporate features with arbitrary data types and multiple predictive models, and (2) propose a heuristic procedure to find the most useful explanations depending on the context. We then contrast counterfactual explanations with methods that explain model predictions by weighting features according to their importance (e.g., SHAP, LIME) and present two fundamental reasons why we should carefully consider whether importance-weight explanations are well-suited to explain system decisions. Specifically, we show that (i) features that have a large importance weight for a model prediction may not affect the corresponding decision, and (ii) importance weights are insufficient to communicate whether and how features influence decisions. We demonstrate this with several concise examples and three detailed case studies that compare the counterfactual approach with SHAP to illustrate various conditions under which counterfactual explanations explain data-driven decisions better than importance weights.