Differential constraints are a class of finite difference equations specified over functions from the powerset of a finite set into the reals. We characterize the implication problem for such constraints in terms of lattice decompositions, and give a sound and complete set of inference rules. We relate differential constraints to a subclass of propositional logic formulas, allowing us to show that the implication problem is coNP-complete. Furthermore, we apply the theory of differential constraints to the problem of concise representations in the frequent itemset problem by linking differential constraints to disjunctive rules. We also establish a connection to relational databases by associating differential constraints to positive boolean dependencies.