In the covariant quantization one is able to get directly (from the constraint $p^\mu_\mu+m^2=0$) the Klein-Gordon equation. But if one uses the parametrization $\tau=X^0$ then the Schrodinger equation is not local anymore (the reason is the square root in the hamiltonian).
I think it has to do with antiparticles. So my guess is that covariant quantization is equivalent to light-cone quantization but it is not equivalent to static gauge quantization.
