Pure rotational spectrum just is a way of distinguishing from rotational-vibrational spectrum. The former is usually in the microwave region of the spectrum and the latter is in the IR. Both kinds of spectroscopy generally give similar information - the difference is that in the ro-vibrational spectra, there are transitions between the vibrational and rotational states simultaneously, whereas in the pure rotational spectrum there are only transitions observed between rotational states within a single vibrational level (because of the low energy involved).
Most molecules in the gaseous state have pure rotational (and ro-vibrational) spectra, although in some cases selection rules come into play. Nonpolar molecules, for example, do not have rotational spectra using direct microwave irradiation because there is no dipole moment change during the transition. To see rotational states of nonpolar molecules, you usually have to use something like Raman spectroscopy, which uses a different set of selection rules. This is why HCl has a very nice pure rotational spectrum but H2 does not.