We compare the spectral properties of 227 Gamma Ray Bursts (GRBs) detected by the Fermi Gamma Ray Burst Monitor (GBM) up to February 2010 with those of bursts detected by the CGRO/BATSE instrument. Out of 227 Fermi GRBs, 166 have a measured peak energy E_peak_obs of their uF( u) spectrum: of these 146 and 20 belong the long and short class, respectively. Fermi long bursts follow the correlations defined by BATSE bursts between their E_peak_obs vs fluence and peak flux: as already shown for the latter ones, these correlations and their slopes do not originate from instrumental selection effects. Fermi/GBM bursts extend such correlations toward lower fluence/peak energy values with respect to BATSE ones whereas no GBM long burst with E_peak_obs exceeding a few MeV is found, despite the possibility of detecting them. Again as for BATSE, $sim$ 5% of long and almost all short GRBs detected by Fermi/GBM are outliers of the E_peak-isotropic equivalent energy (Amati) correlation while no outlier (neither long nor short) of the E_peak-isotropic equivalent luminosity (Yonetoku) correlation is found. Fermi long bursts have similar typical values of E_peak_obs but a harder low energy spectral index with respect to all BATSE events, exacerbating the inconsistency with the limiting slopes of the simplest synchrotron emission models. Although the short GRBs detected by Fermi are still only a few, we confirm that their E_peak_obs is greater and the low energy spectrum is harder than those of long ones. We discuss the robustness of these results with respect to observational biases induced by the differences between the GBM and BATSE instruments.
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