Policy in the two halves of the Roman Empire became more pronounced as time went by and their differences became more evident.
After the initial split there was a natural assumption that the Empire was a body one and the same as always, but administratively it needed to rulers to better enable it to deal with increasing uncertainty, greater threats and declining economic power. The two Empires therefore cooperated and their political structure was strongly related by marriage and family connection.
The Battle of Adrianople in 376 marked a point of change. As usual the two Empires acted cooperatively, the Western Emperor sending his army to aid his Eastern colleague. The defeat of the East Romans threw the military situation in the Balkans in to utter disarray. The ERE had been fairly sufficient in maintenance of its Persian frontier, but Adrianople created a severe crisis. This diminished her ability to act in concert with the WRE, she was simply too busy dealing with the Goths. As a result the WRE had to continue her struggles against the Germans on her own.
At the death of Theodosius in 395 the two Empires drifted further apart. Though the two royal families were strongly connected to eachother, the Emperors themselves were weak and effective power was wielded by court favourites. These court favourites were now actively intriguing against the court of their sister empire. Some have suggested that WRE general Stilicho actually acted with the cooperation of Alaric the Ostrogoth against the ERE.
The problems in the Balkans left the ERE grappling with military threats and the problem of creating a reliable national army to replace foederati mercenaries. This was a problem they would continue grappling with for centuries, while continually needing to hold the Eastern frontier. The WRE, now unable to rely on the East for consistent help, now fought a losing battle just to survive. Every now and then an ERE army would turn up to try and enforce Roman authority, such as to install Valentinian III or Basilacius' campaign against the Vandals. But essentially the two Empires grew apart. The ERE battled to adapt to a changing world and reinvent itself as a viable state with a reliable military. The WRE simply did whatever it could to stave off death, but its situation was already terminal.