With the aim of investigating the binary population of the 700 Myr old Praesepe cluster, we have observed 149 G and K-type cluster members using adaptive optics. We detected 26 binary systems with an angular separation ranging from less than 0.08 to 3.3 arcsec (15-600 AU). After correcting for detection biases, we derive a binary frequency (BF) in the logP (days) range from 4.4 to 6.9 of 25.3 +/- 5.4%, which is similar to that of field G-type dwarfs (23.8%, Duquennoy & Mayor 1991). This result, complemented by similar ones obtained for the 2 Myr old star forming cluster IC 348 (Paper II) and the 120 Myr old Pleiades open cluster (Paper I), indicates that the fraction of long-period binaries does not significantly evolve over the lifetime of galactic open clusters. We compare the distribution of cluster binaries to the binary populations of star forming regions, most notably Orion and Taurus, to critically review current ideas regarding the binary formation process. We conclude that it is still unclear whether the lower binary fraction observed in young clusters compared to T associations is purely the result of the early dynamical disruption of primordial binaries in dense clusters or whether it reflects intrinsically different modes of star formation in clusters and associations. We also note that if Taurus binaries result from the dynamical decay of small-N protostellar aggregates, one would predict the existence of a yet to be found dispersed population of mostly single substellar objects in the Taurus cloud.