  The standard model coherent elastic neutrino-nucleus scattering (CE$\nu$NS)
cross section is subject to nuclear form factor uncertainties, mainly driven by
the root-mean-square radius of the neutron density distribution. Motivated by
COHERENT phases I-III and future multi-ton direct detection dark matter
searches, we evaluate these uncertainties in cesium iodide, germanium, xenon
and argon detectors. We find that the uncertainties become relevant for
momentum transfers $q\gtrsim 20$ MeV and are essentially independent of the
form factor parameterization. Consequently, form factor uncertainties are not
important for CE$\nu$NS induced by reactor or solar neutrinos. Taking into
account these uncertainties, we then evaluate their impact on measurements of
CE$\nu$NS at COHERENT, the diffuse supernova background (DSNB) neutrinos and
sub-GeV atmospheric neutrinos. We also calculate the relative uncertainties in
the number of COHERENT events for different nuclei as a function of recoil
energy. For DSNB and atmospheric neutrinos, event rates at a liquid argon
detector can be uncertain to more than 5%. Finally, we consider the impact of
form factor uncertainties on searches for nonstandard neutrino interactions,
sterile neutrinos and neutrino generalized interactions. We point out that
studies of new physics using CE$\nu$NS data are affected by neutron form factor
uncertainties, which if not properly taken into account may lead to the
misidentification of new physics signals. The uncertainties quantified here are
also relevant for dark matter direct detection searches.
