In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum- or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples.