The interparticle interactions in colloidal suspensions of charged disks of Laponite clay in water were investigated using dynamic light scattering (DLS) and Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. We studied the effects of clay concentration (C-L), the concentration of externally added salt (CS), and temperature (T) on the microscopic dynamics of the clay suspensions. The fast (tau(1)) and mean slow relaxation times (tau(ww)) of Laponite suspensions were extracted from intensity autocorrelation functions measured at different waiting times (t(w)) after sample preparation. Comprehensive Laponite concentration-salt concentration-temperature-time superpositions of both the microscopic diffusive time scales and the stretching exponent corresponding to the slow relaxation process highlight the self-similar nature of the energy landscapes of the Laponite suspensions. The evolution of the sodium ion concentration in the aging suspension with t(w), measured for several values of C-L, C-S, and T, was used in a DLVO analysis of the free energy of the suspension for two charged disks parallely approaching one another. This analysis confirms that, in addition to repulsive interparticle interactions, attractive interactions also play a pivotal role in the microscopic dynamics of spontaneously evolving Laponite suspensions.