  The differential cross section for the process $\gamma \gamma \to \pi^0
\pi^0$ has been measured in the kinematic range 0.6 GeV $< W < 4.1$ GeV, $|\cos
\theta^*|<0.8$ in energy and pion scattering angle, respectively, in the
$\gamma\gamma$ center-of-mass system. The results are based on a 223 fb$^{-1}$
data sample collected with the Belle detector at the KEKB $e^+ e^-$ collider.
Using the data with $W > 1.4$ GeV, we obtain results on light-quark resonances
and charmonia. We also compare the observed angular dependence and ratios of
cross sections for neutral-pair and charged-pair production to QCD models.
Differential cross sections are fitted in the energy region, $1.4 \GeV < W <
2.2 \GeV$, with a simple model where partial waves consist of resonances such
as $f_2'(1525)$, $f_2(1950)$ and $f_4(2050)$ and smooth backgrounds. In the
higher energy region, we observe production of the $\chi_{c0}$ charmonium state
and obtain the product of its two-photon decay width and the branching fraction
to $\pi^0\pi^0$. The energy and angular dependences above 3.1 GeV are
compatible with those measured in the $\pi^+\pi^-$ channel, and in addition we
find that the cross section ratio, $\sigma(\pi^0\pi^0)/\sigma(\pi^+\pi^-)$, is
$0.32 \pm 0.03 \pm 0.05$ on average in the 3.1-4.1 GeV region.
