The differential cross sections for the process gamma gamma -> pi(0)pi(0) have been measured in the kinematic range 0.6 GeV < W < 4.1 GeV, vertical bar cos theta* vertical bar < 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 +/- 0.03 +/- 0.05 on average in the 3.1-4.1 GeV region.