A western "strongman" is replaced by an eastern one. Paradoxically it might be better for the WRE quite possibly Maxentius would not move the main capital from Rome and if he did certainly not to Byzantium. "Maxentinopolis" the eastern administrative centre would either be Antioch or Alexandria.
How on Earth is Maxentius conquering the east? He just barely was able to hold his own against Constantine (and besides, he still has a lot of other problems to deal with, namely his unpopularity) and the winner of the Licinius vs. Maximinius Daia conflict is going to be much stronger than him. Licinius held his own well in his battles with Constantine, and Constantine was far and away a better general than Maxentius.
Also, wasn't Maxentius allied to one of them? EDIT: He was allied to Daia. Either way, unlike Constantine, it never seemed like he had the ability or the urge to take over sole rule of the empire. If anything, Maxentius is a ripe target for either Licinius or Daia. A
Assuming he does win though, there's no reason he's not going to use Nikomedia and Antioch, like previous emperors. Alexandria really is too far removed from everything.
Anyway, interesting perspective from historians looking back on it: they would wonder why Constantine chose to invade Italy when Galerius had proven what a bad idea that was, rather than remain in Gaul and let Maxentius come to him.
EDIT: It would be interesting to see how Christianity would develop too, without official state backing and thus no Nicene Creed.