I think that even geographically and climatically these two regions were put at odds with each other too. Carthage and Rome.
If you look at the map Muslims occupy mostly desert, semi-desert areas whereas West is completely green valleys, although green is the colour much liked in deserts too.
The games, ludi, had a religious element to them during early Rome in honor of Jupiter Optimus Maximus. This ended after 220 BC, and while other types of festivals, such as chariot races, still maintained a religious facet, the games did not.
So, starting from about 220 BC secular games became really secular, without any religious meaning. From that period till the time when prophet Jesus said that any man should give God's to God and give Caesar's to Caesar the fate of separation of secular from religious has been sealed for Christian West.
I don't see that division of secular from religion in Muslim world of today, let alone times of early Caliphates.