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In this paper the author compares behaviors of systems which can be describedby fractional differential and fractional difference equations using thefractional and fractional difference Caputo Standard $\alpha$-Families of Mapsas examples. The author shows that properties of fractional difference maps(systems with falling factorial-law memory) are similar to the properties offractional maps (systems with power-law memory). The similarities (types ofattractors, power-law convergence of trajectories, existence of cascade ofbifurcations and intermittent cascade of bifurcations type trajectories, anddependence of properties on the memory parameter $\alpha$) and differences inproperties of falling factorial- and power-law memory maps are investigated.

Caputo Standard $\alpha$-Family of Maps: Fractional Difference vs. Fractional