The ratio between the prompt psi (2S) and J/psi yields, reconstructed via their decays into mu(-)mu(-) is measured in Pb-Pb and p-p collisions at root sNN = 2.76 TeV. The analysis is based on Pb-Pb and p-p data samples collected by CMS at the Large Hadron Collider, corresponding to integrated luminosities of 150 mu b(-1) and 5.4 mu b(-1), respectively. The double ratio of measured yields (N phi((2S))/NJ/(Pb-Pb)/N-psi(2S)/NJ/psi)(p-p) is computed in three Pb-Pb collision centrality bins and two kinematic ranges: one at midrapidity, vertical bar y vertical bar < 1.6, covering the transverse momentum range 6.5 < pT < 30 GeV= c, and the other at forward rapidity, 1.6 < vertical bar y vertical bar < 2.4, extending to lower pT values, 3 < pT < 30 GeV/c. The centralityintegrated double ratio changes from 0.45 +/- 0.13(stat) +/- 0.07dsyst_ in the first range to 1.67 +/- 0.34(stat) +/- 0.27dsyst_ in the second. This difference is most pronounced in the most central collisions.