  Two self-consistent schemes involving Hedin's $GW$ approximation are studied
for a set of sixteen different atoms and small molecules. We compare results
from the fully self-consistent $GW$ approximation (SC$GW$) and the
quasi-particle self-consistent $GW$ approximation (QS$GW$) within the same
numerical framework. Core and valence electrons are treated on an equal footing
in all the steps of the calculation. We use basis sets of localized functions
to handle the space dependence of quantities and spectral functions to deal
with their frequency dependence. We compare SC$GW$ and QS$GW$ on a qualitative
level by comparing the computed densities of states (DOS). To judge their
relative merit on a quantitative level, we compare their vertical ionization
potentials (IPs) with those obtained from coupled-cluster calculations CCSD(T).
Our results are futher compared with "one-shot" $G_0W_0$ calculations starting
from Hartree-Fock solutions ($G_0W_0$-HF). Both self-consistent $GW$ approaches
behave quite similarly. Averaging over all the studied molecules, both methods
show only a small improvement (somewhat larger for SC$GW$) of the calculated
IPs with respect to $G_0W_0$-HF results. Interestingly, SC$GW$ and QS$GW$
calculations tend to deviate in opposite directions with respect to CCSD(T)
results. SC$GW$ systematically underestimates the IPs, while QS$GW$ tends to
overestimate them. $G_0W_0$-HF produces results which are surprisingly close to
QS$GW$ calculations both for the DOS and for the numerical values of the IPs.
