Today, most people in North Africa from Egypt to Morocco are Arabic in culture and language, but to the east of the Arabian Peninsula, even in places conquered by the early caliphates like Persia, the local culture was not replaced nearly as thoroughly. Why was North Africa more heavily assimilated into Arab culture than places like Persia and the west of the Indian subcontinent?
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Nicholas Ostler, in Empires of the word: a language history of the world, points out that, in general, the spread of languages (at the expense of others) is fostered by, specially, similarities between the replaced and the replacement and by the virtue of being the language of the first literary culture introduced in the region.
Take Latin, for example: it spread to encompass practically all of Italy and almost all of Gaul and Hispania, which were then peopled mainly by speakers of Indo-European languages, only the language isolate Basque surviving around the western end of the Pyrenees.
Of course there were some exceptions to the rule (which arguably prove the rule): Etruscan, another non-Indo-European language vanished, but it was right next to Latium and surrounded by other Italic languages, so it might have been absorbed due to sheer numbers. In Celtic-speaking Britannia, Latin also disappeared, probably because it was a fairly marginal province in the first place and also because (apparently) it suffered massively from plague and migration.
In North Africa west of Egypt, Latin didn't spread much beyond the coast: even there it faced Punic, the Semitic language of the conquered Carthaginians, so, a tongue that had its own literary culture and belonged to another language family (Afro-Asiatic). Take the Emperor Septimius Severus for example: a native of Leptis Magna (near modern Tripoli, in Libya), who reigned from 193 to 211 (so more than three centuries after Carthage got salted the end of the Third Punic War), he wasn't a native Latin speaker (he even was ashamed of his sister's broken Latin!). Further inland, it faced Berber languages, which, although hadn't literary cultures of their own, also belonged to the Afro-Asiatic family.
In the Eastern Empire, Latin only spread to the Northern Balkans; elsewhere, Greek, another Indo-European language, prevailed as the language of high culture, to the point that even the Romans themselves looked up to it, many of the elite having learned it as second language (the Emperor Claudius is reported to have spoken of Latin and Greek as "our two languages"). Beyond being the prestige language of the Eastern Mediterranean par excellence, Greek was also the majority (or at least having a large minority) language in Greece itself and the coasts of Anatolia (the Asian portion of modern Turkey) and many Hellenistic cities founded in Egypt (like Alexandria) and the Fertile Crescent (like Antioch) by Alexander the Great of Macedon and his successor states.
Even in inland Anatolia Greek would eventually replace other local Indo-European languages, specially when Christian missionaries used Greek, now with additional prestige due to being the original language in which the Gospels were written, to spread the Christian faith. Apparently a similar process happened with Latin in Gaul and Hispania in the last centuries of the Western Roman Empire, which until then, or perhaps even after the fall (!!!) still had the majority/plurality of the population speaking local tongues. For this, see Language birth and death by Salikoko S. Mufwene and The linguistic situation in the western provinces of the Roman Empire by Edgar C. Polomé.
Greek, however, didn't spread much to the masses of Egypt, the Levant and Mesopotamia, for there it faced languages with very ancient literary cultures. In the land of the Nile, there was Coptic (really Egyptian written in an alphabet derived from the Greek one), an Afro-Asiatic language, remained the tongue of the majority. Greek being dominant only in Hellenistic cities, while Latin probably didn't even stand a chance there, being a language of the tiny Roman bureaucracy. In the Levant and Mesopotamia the story was similar, with Semitic, Afro-Asiatic Aramaic in the place of Coptic.
Finally, to the East of Mesopotamia, we reach Persia (or rather, Iran; Persia being a region in Iran) and India, areas where Indo-European languages were spoken and, like Greek and Latin, had ancient literary cultures of their own. After the decline of the Hellenistic Seleucid Empire, a resurgent Iran, led by the Arsacid and Sassanid dynasties, progressively (although not always consistently) stamped out Greek influence (which was stronger in the Mediterranean coast anyway).
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