The nuclear modification factor RAA and the azimuthal anisotropy coefficient v[2] of prompt and nonprompt (i.e. those from decays of b hadrons) J/psi mesons, measured from PbPb and pp collisions at sqrt(s[NN]) = 2.76 TeV at the LHC, are reported. The results are presented in several event centrality intervals and several kinematic regions, for transverse momenta pt > 6.5 GeV/c and rapidity abs(y) < 2.4, extending down to pt = 3 GeV/c in the 1.6 < abs(y) < 2.4 range. The v[2] of prompt J/psi is found to be nonzero, but with no strong dependence on centrality, rapidity, or pt over the full kinematic range studied. The measured v[2] of nonprompt J/psi is consistent with zero. The RAA of prompt J/psi exhibits a suppression that increases from peripheral to central collisions but does not vary as a function of either y or pt in the fiducial range. The nonprompt J/psi RAA shows a suppression which becomes stronger as rapidity or pt increase. The v[2] and nuclear suppression of open and hidden charm, and of open charm and beauty, are compared.