The authors consider the equivalence problems for second order linear difference equations associated with the operators \[ {\mathcal L}:=a_n{\mathbf S}^2+b_n{\mathbf S}+c_n, \quad \widehat{{\mathcal L}}:=\widehat{a}_n{\mathbf S}^2+\widehat{b}_n{\mathbf S}+\widehat{c}_n, \] where \({\mathbf S}\) is the forward shift operator \({\mathbf S}(u_n)=u_{{\mathbf S}(n)}=u_{n+1}\), in the context of the theory of equivariant moving frames [M. Fels and P. J. Olver, Acta Appl. Math. 55, No. 2, 127–208 (1999; Zbl 0937.53013)]. The authors provide criteria for the direct equivalence problem \[ {\mathcal L}[u_n]=\widehat{{\mathcal L}}[\widehat{u}_n], \] the gauge (or eigenvalue) equivalence problem \[ {\mathcal L}[u_n]=\lambda u_n, \quad \widehat{{\mathcal L}}[\widehat{u}_n]=\lambda\widehat{u}_n, \] and the projective equivalence problem \[ {\mathcal L}[u_n]=0, \quad \widehat{{\mathcal L}}[\widehat{u}_n]=0. \]