The Gamma-ray Burst Monitor (GBM) on board Fermi allows us to study the spectra of gamma-ray bursts (GRBs) over an unprecedented wide energy range (8 keV-35 MeV). We compare the spectral properties of short and long GRBs detected by the GBM (up to 2010 March) with those of GRBs detected by the Burst And Transient Source Experiment (BATSE) on board the Compton Gamma Ray Observatory (CGRO). GBM and BATSE long bursts have similar distributions of fluence (F), E obs peak and peak flux (P) but GBM bursts have a slightly harder low-energy spectral index α with respect to BATSE GRBs. GBM and BATSE short bursts have similar distributions of fluence, α and peak flux, with GBM bursts having slightly larger E obs peak. We discuss these properties in light of the correlations found between E obs peak and the fluence and the peak flux. GBM bursts confirm that these correlations are not determined by instrumental selection effects. Indeed, GBM bursts extend the E obs peak-F and E obs peak-P correlations both in fluence/peak flux and in peak energy. No GBM long burst with E obs peak exceeding a few MeV is found, despite the possibility of detecting it. Similarly to what is found with BATSE, there are 3 per cent of GBM long bursts (and almost all short ones) that are outliers at more than 3σ of the E peak-E iso correlation. In contrast, there is no outlier of the E peak-L iso correlation, for both long and short GBM bursts.