Two fundamental modelling problems in nonlinear discrete-time control systems are studied using the language of differential forms. The discrete-time nonlinear single-input single-output systems to be studied are described by input-output (i/o) difference equations, i.e. a high order difference equation relating the input, the output and a finite number of their time shifts. A new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case. Our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous-time systems. The second problem to be addressed is the realization problem. The i/o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization. Necessary and sufficient conditions are given for the existence of a (local) state-space realization of the irreducible i/o difference equation. These conditions are formulated in terms of the integrability of certain subspaces of one-forms, classified according to their relative degree. The sufficiency part of the proof gives a constructive procedure (up to finding the integrating factors and integration of the set of one-forms) for obtaining a locally observable and accessible state-space system. If the system is not in the irreducible form, one has first to apply the reduction procedure to transform the system into the irreducible form.