The difference can be viewed in two ways, I guess.
Firstly, for relaxation spectroscopy, you apply a forced oscillation on your sample. That can be mechanical (dynamic rheology, mHz to .1kHz, several kHz with very special, not commerically available equipment) or electrical (dielectric spectroscopy, low mHz to GHz range). The mathematical concepts behind both methods are very similar. For vibrational, MR, UV, etc. spectroscopy, the sample does just not interact with frequencies outside of the linewidth of some QM transition. The width of a resonant peak is given by the energy needed for the transition, and the lifetime of the excited state. The width of a feature in a relaxation spectrum is typically at least half an order of magnitude, from "sample can follow the external stimulus and stay close to a (dynamic) equillibrium" to "external stimulus is much too fast for the sample to follow at all".
The other point is that in relaxation spectroscopy, energy is transferred and dissipated into the sample below a specific frequency (invariably as heat), and reflected (elastically) at higher frequencies. The energy is not quantised, because the induced change in the sample is translatory. The peak frequency itself has no real significance, it's inverse is called a "relaxation time" of the sample. A statistical quantity, mostly. For QM transitions, energy is only transmitted at the given frequency/energy, and nothing happens otherwise, except Rayleigh scattering.
Basically, resonant spectroscopy deals with individual quantum mechanical species, and relaxation spectroscopy deals with the classical, unquantised newtonian properties of the whole sample. NMR is a bit of a hybrid: The induced transitions are purest QM, but you can only observe the classic induction of the ensemble, never individual photons.
Technically, relaxation spectra are usually taken by applying the stimulus at discrete frequencies, stepwise, and you want to have a phase-sensitive detector (voltage - current, deformation - torque).