  The differential cross sections for the process $\gamma \gamma \to \pi^0
\pi^0$ have 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.
The differential cross sections are fitted in the energy region 1.7 GeV $< W <$
2.5 GeV to confirm the two-photon production of two pions in the G wave. 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 branching fraction to
$\pi^0\pi^0$. We also compare the observed angular dependence and ratios of
cross sections for neutral-pion and charged-pion pair production to QCD models.
The energy and angular dependence 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.
