"The empire, long divided, must unite; long united, must divide. Thus it has ever been." It must be remembered that, until the fall of the Western Empire, the division was officially an administrative one:
de iure there is only
One Roman Empire with
Two emperors. What I imagine that is going to happen is a "China-like" continuation of a cycle of unity-disunity, with the more ambitious emperors dreaming of achieving the same level of domination over the entire roman world as Constantine or Theodosius did, and with the Empire fragmenting after the force of reality imposes the fact that the Empire as it is, is too big for one single ruler. That is, until the two halves run out of resources to dominate each other. Knowing that the East is usually richer and more powerful than the West, the most plausible result is for the East conquering or puppeteering the West, unless a lucky or better in the art of war Western Emperor is able to defeat the East. Of course, it also depends on the extent of the Empires: The situation is different if the West is reduced to Italy or/and if the East is reduced to Basil II´s frontiers. Also, we also must remember the religious aspect, with the Emperors fighting over schismatic decisions, taken by the patriarchs under their rule, as well as the ambition of certain Patriarchs (like the one of Rome) to exercise Primacy/superiority over the entire Church, which would surely receive support from an emperor and vice versa. About colonies, the East is rich enough and with full control of about spices and Asian trade to be interested in colonies,specially American ones, for which also does not have the correct geographical location. Only the West can be interested in such ventures, and you need it to:
- Having coast near the Atlantic
- Having such a bad relationship with whoever dominates the commercial routes in the East that motivates the search for a route through the West, finding the New World in the process.
Regarding the language or system of government, that also depends on what state the Empires have survived. While it is possible or worthwhile to maintain the fiction that the two empires are one, the East will maintain Latin among its official languages. Of the two, the West is the most exposed to suffer a feudalizing phenomenon, with the consequent decentralization, compared to the East, which, as demonstrated by OTL, can maintain high levels of Autocracy and central power even if they are in their last legs.