We propose a one-loop induced radiative neutrino mass model with anomaly free flavour dependent gauge symmetry: $mu$ minus $tau$ symmetry $U(1)_{mu-tau}$. A neutrino mass matrix satisfying current experimental data can be obtained by introducing a weak isospin singlet scalar boson that breaks $U(1)_{mu-tau}$ symmetry, an inert doublet scalar field, and three right-handed neutrinos in addition to the fields in the standard model. We find that a characteristic structure appears in the neutrino mass matrix: two-zero texture form which predicts three non-zero neutrino masses and three non-zero CP-phases from five well measured experimental inputs of two squared mass differences and three mixing angles. Furthermore, it is clarified that only the inverted mass hierarchy is allowed in our model. In a favored parameter set from the neutrino sector, the discrepancy in the muon anomalous magnetic moment between the experimental data and the the standard model prediction can be explained by the additional neutral gauge boson loop contribution with mass of order 100 MeV and new gauge coupling of order $10^{-3}$.
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