Simultaneous estimation will surely produce different output than stepwise estimation. The conditional mean model is estimated assuming a GARCH-type conditional variance in the simultaneous case but a constant variance in the stepwise case, yielding different optima. The conditional variance model is estimated in line with the first optimum in the simultaneous case but in line with the second optimum in the stepwise case, again yielding different results in the two cases. There is no guarantee you will get "nicer" results (more significant coefficients and/or better behaved standardized errors) form simultaneous estimation, but conceptually this is more sensible approach as it avoids making conflicting assumptions at the different stages (unlike the two-step approach). See some related threads where this has been discussed before.
In terms of computational efficiency, I think stepwise estimation could be preferred. In stepwise estimation, fewer parameters are being optimized at a time, so the optimization task is easier and convergence is achieve with less trouble. Of course, there are two stages, so we have two easy tasks instead of a single, difficult one, and thus it could probably go either way.