  We present a detailed study of the energetics of water clusters (H$_2$O)$_n$
with $n \le 6$, comparing diffusion Monte Carlo (DMC) and approximate density
functional theory (DFT) with well converged coupled-cluster benchmarks. We use
the many-body decomposition of the total energy to classify the errors of DMC
and DFT into 1-body, 2-body and beyond-2-body components. Using both
equilibrium cluster configurations and thermal ensembles of configurations, we
find DMC to be uniformly much more accurate than DFT, partly because some of
the approximate functionals give poor 1-body distortion energies. Even when
these are corrected, DFT remains considerably less accurate than DMC. When both
1- and 2-body errors of DFT are corrected, some functionals compete in accuracy
with DMC; however, other functionals remain worse, showing that they suffer
from significant beyond-2-body errors. Combining the evidence presented here
with the recently demonstrated high accuracy of DMC for ice structures, we
suggest how DMC can now be used to provide benchmarks for larger clusters and
for bulk liquid water.
