  In this paper the author compares behaviors of systems which can be described
by fractional differential and fractional difference equations using the
fractional and fractional difference Caputo Standard $\alpha$-Families of Maps
as examples. The author shows that properties of fractional difference maps
(systems with falling factorial-law memory) are similar to the properties of
fractional maps (systems with power-law memory). The similarities (types of
attractors, power-law convergence of trajectories, existence of cascade of
bifurcations and intermittent cascade of bifurcations type trajectories, and
dependence of properties on the memory parameter $\alpha$) and differences in
properties of falling factorial- and power-law memory maps are investigated.
