Differential measurements of the elliptic (v(2)) and hexadecapole (v(4)) Fourier flow coefficients are reported for charged hadrons as a function of transverse momentum (p(T)) and collision centrality or number of participant nucleons (N(part)) for Au+Au collisions at sq.rt(s(NN))=200 GeV. The v(2,4) measurements at pseudorapidity |η|≤0.35, obtained with four separate reaction-plane detectors positioned in the range 1.0<|η|<3.9, show good agreement, indicating the absence of significant Δη-dependent nonflow correlations. Sizable values for v(4)(p(T)) are observed with a ratio v(4)(p(T),N(part))/v(2)(2)(p(T),N(part))≈0.8 for 50≲N(part)≲200, which is compatible with the combined effects of a finite viscosity and initial eccentricity fluctuations. For N(part)≳200 this ratio increases up to 1.7 in the most central collisions.