Fitting ARMA-GARCH sequentially vs simultaneously

I am interested in fitting an ARMA-GARCH model to my data. After reading a few pages online I did so sequentially by first applying ARMA and then feeding the residuals into GARCH. I then took the estimated mean from ARMA and variance from GARCH to construct a forecast. However, I have not achieved too great of a result and after some further reading it seems this method is acceptable though not ideal. Others suggest the ARMA-GARCH parameters should be instead fit simultaneously.

Will doing so result in a different output or is it a matter of computational efficiency?