Especially in their military, politic, religion, etc. How does having Persia in their east affect the Eastern Rome, and how does having a lot of rebellions affect the Western one? (I don't recall any faction the latter can call "rival")
Also how do they behave with each other? Are they friendly? Enemies? Frenemies?
When the second major split occured in 395 AD,after the death of Theodosius I the Great,his 2 sons,Arcadius and Honorius, inherited the two empires.Altough they weren't in a formal alliance ,they pretty much swore to help each othe rin times of need,altough this wouldn't happen often.The religion was mostly the same(christian,being made the only legal religion by Constantine the Great after a dream in which it was said he would achieve victory if his soldiers worshiped the cross),and by all accounts the Eastern Roman Empire was more developed,due to the fact that it didn't require to spend more on defending itself from migratory invasions.In terms of army size,the Western Roman Empire maintained a larger army that the Eastern Roman Empire,which later led to mutinies and more migratory invasions.Altough the West fell in 476 AD ,after decades of struggle,the East continued to prosper and even reconquered Rome in the 6th century.