  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 \nuF(\nu) 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.
