The time discretization of stochastic spectral fractional wave equation is studied by using the difference methods. Firstly, we exploit rectangle formula to get a low order time discretization, whose the strong convergence order is smaller than $1$ in the sense of mean-squared $L^2$-norm. Meanwhile, by modifying the low order method with trapezoidal rule, the convergence rate is improved at expenses of requiring some extra temporal regularity to the solution. The modified scheme has superlinear convergence rate under the mean-squared $L^2$-norm. Several numerical experiments are provided to confirm the theoretical error estimates.