This paper provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by $ \dot{F}^s_{p,q}(\mathbb R^n) $ and $ \dot{B}^s_{p,q}(\mathbb R^n) $ respectively, in terms of maximal functions of the mean values of iterated difference. It also furnishes the reader with inequalities in $ \dot{F}^s_{p,q}(\mathbb R^n) $ in terms of iterated difference and in terms of iterated difference along coordinate axes. The corresponding inequalities in $ \dot{B}^s_{p,q}(\mathbb R^n) $ in terms of iterated difference and in terms of iterated difference along coordinate axes are also considered. The techniques used in this paper are of Fourier analytic nature and the Hardy-Littlewood and Peetre-Fefferman-Stein maximal functions.