  Total cross sections, angular distributions, and invariant-mass distributions
have been measured for the photoproduction of $\pi^0\pi^0$ pairs off free
protons and off nucleons bound in the deuteron. The experiments were performed
at the MAMI accelerator facility in Mainz using the Glasgow photon tagging
spectrometer and the Crystal Ball/TAPS detector. The accelerator delivered
electron beams of 1508 and 1557~MeV, which produced bremsstrahlung in thin
radiator foils. The tagged photon beam covered energies up to 1400~MeV. The
data from the free proton target are in good agreement with previous
measurements and were only used to test the analysis procedures. The results
for differential cross sections (angular distributions and invariant-mass
distributions) for free and quasi-free protons are almost identical in shape,
but differ in absolute magnitude up to 15\%. Thus, moderate final-state
interaction effects are present. The data for quasi-free neutrons are similar
to the proton data in the second resonance region (final state invariant masses
up to $\approx$1550~MeV), where both reactions are dominated by the
$N(1520)3/2^-\rightarrow \Delta(1232)3/2^+\pi$ decay. At higher energies,
angular and invariant-mass distributions are different. A simple analysis of
the shapes of the invariant-mass distributions in the third resonance region is
consistent with strong contributions of an $N^{\star}\rightarrow N\sigma$ decay
for the proton, while the reaction is dominated by a sequential decay via a
$\Delta\pi$ intermediate state for the neutron. The data are compared to
predictions from the Two-Pion-MAID model and the Bonn-Gatchina coupled channel
analysis.
