Here are some of my thoughts on how each empire would evolve.
The Western Empire will remain under the rule of the Latin elite, from Rome, and will probably have a succession of Emperors similar to the OTL Julio-Claudian dynasty. In the short term, W-Rome will finish conquering Hispania. With no concerns about overextending himself, Augustus and his successors may concentrate more on conquering Britain, and maybe Germania. I'm not sure how that would go. Agriculturally, the empire would be very sharply split north to south. Home style agriculture will not do as well in northern Gaul or Germania, and the Germans will be hard to control, being so spread out. OTOH, the area is ripe for colonization. It is so empty. Overtime, the Roman Legions will come to be made more and more of Germanic and Celtic people, until the Latin elite is overthrown and replaced by a romanized German elite
The Eastern Empire has a majority Greek and Hellenic population, and so will become increasingly Greek in culture overtime, looking on themselves as the natural successors to the glorious reign of Alexander the Great. Mark Anthony and Cleopatra will likely go through with their plan to have Caesarion marry Antonia, and their dynasty will continue to rule the east and peacefully annex Egypt. The Eastern Romans will likely maintain the capital at Alexandria, the largest and richest city in the east. Athens was long past it's glory by this point. Militarily, they will spend nearly all their time fighting with the Parthian Empire in Persia, though they may also conquer Dacia, as OTL Rome did under Trajan. It will be in Eastern Rome that Christianity will first rise, though friendly relations between the empires will mean that it will probably spread just as far as in OTL.
Southern Rome will (most unwillingly) come to be seen as the heirs to the legacy of Carthage. Their culture will become more Punic overtime, as that was the majority of the population in that region. Initially, they will concentrate on conquering Mauritania. Afterward, with no other directions form expansion, they will concentrate on re-tracing the voyages of Hanno the Navigator around Africa, in search of new lands and trading partners. Eventually, by sea or by camel overland, they will make contact with the Sahel civilizations of West Africa. There will be a great deal of trade (maybe even some fighting, the Moroccans IOTL invaded this region several times in the middle ages) which will lead to a lot of cultural exchange between these regions. This will accelerate the development of West Africa, and also cause Southern Rome to take on more elements of African culture.