  As numerical calculations of inspiralling neutron-star binaries reach values
of accuracy that are comparable with those of binary black holes, a fine
budgeting of the various sources of error becomes increasingly important. Among
such sources, the initial data is normally not accounted for, the rationale
being that the error on the initial spacelike hypersurface is always far
smaller than the one gained during the evolution. We here consider critically
this assumption and perform a comparative analysis of the gravitational
waveforms relative to essentially the same physical binary configuration when
computed with two different initial-data codes, and then evolved with the same
evolution code. More specifically, we consider the evolution of irrotational
neutron-star binaries computed either with the pseudo-spectral code \lorene{},
or with the newly developed finite-difference code \cocal{}; both sets of
initial data are subsequently evolved with the high-order evolution code
\whiskythc{}. In this way we find that despite the initial data shows global
(local) differences that are $\lesssim 0.02\%\ (1\%)$, the gravitational-wave
phase at the merger time differs by $\sim 0.5$ radians after $\sim 3$ orbits, a
surprisingly large value. Our results highlight the highly nonlinear impact
that errors in the initial data can have on the subsequent evolution and the
importance of using exactly the same initial data when comparative studies are
done.
