A sample of 47 faint Gigahertz Peaked Spectrum (GPS) radio sources selected from the Westerbork Northern Sky Survey (WENSS) has been imaged in the optical and near-infrared, resulting in an identification fraction of 87 per cent. The R-IR-K colours of the faint optical counterparts are as expected for passively evolving elliptical galaxies, assuming that they follow the R-band Hubble diagram as determined for radio-bright GPS galaxies. We find evidence that the radio spectral properties of the GPS quasars are different from those of GPS galaxies. The observed distribution of radio spectral peak frequencies for GPS sources optically identified with bright stellar objects (presumably quasars) is shifted compared with GPS sources identified with faint or extended optical objects (presumably galaxies), in the sense that a GPS quasar is likely to have a higher peak frequency than a GPS galaxy. This means that the true peak frequency distribution is different for the GPS galaxies and quasars, because the sample selection effects are independent of optical identification. The correlation between peak frequency and redshift that has been suggested for bright sources has not been found in this sample; no correlation exists between R magnitude (and therefore redshift) and peak frequency for the GPS galaxies. We therefore believe that the claimed correlation is actually caused by the dependence of the peak frequency on optical host, because the GPS galaxies are generally at lower redshifts than the quasars. The difference in the peak frequency distributions of the GPS galaxies and quasars is further evidence against the hypothesis that they form a single class of object.