I'm a 'butterfly extremist', so I doubt there would be anything approaching OTL Islam in this timeline, but ignoring that, if something like Islam were to arise, it would indeed be very similar to Manichaeism (though with a more strict monotheistic twist).
With the grassroots popularity of Christianity in North Africa, I doubt the Manichaen emperors would be able to entirely stomp it out. They would undoubtedly try, but the entrenchment of the faith amongst local leaders might prevent total extirpation.
This would lead to some interesting developments once *Christian Persia* arises; they may desire their holy lands (Palestine and Syria), and see North Africa (with its large Christian population) as an opportunity for expansion. If they managed to secure both, even for a short period of time, it might lead to a decisive break between European Rome and their African lands. One could imagine Manichaeism (largely confined to Europe over time) as being synonymous with the continent, whilst Christianity would be considered the barbarous practice of Easterners and Africans.
Unlike OTLs Christian/Islam split, however, the Europeans would be the ones praying four-times a day, and with the religious concerns over eating 'living matter', there might be a higher preponderance of vegetarianism in Europe.
Actually, there might be a number of interesting developments if Manichaeism becomes localised to Europe.
- Buddhist beliefs (which were present IOTL) might be more widely known (reinforcing the vegetarian point), and interest in Asia-beyond-Persia would be high, especially if Manichean communities continue to exist in Central Asia and China. Prester John might just be a Manichean in this timeline.
- Manichean beliefs about the Sun and Moon (e.g. the Moon being an emanation of Jesus) might lead to cultural activities during periods of solstice, lunar eclipses, etc. In Scandinavia, the Midnight Sun might be seen as one of the holiest periods of the year.
- Manichean beliefs about proselytising to 'all nations in all tongues' might preclude the *Roman Church* from centralising around Latin as the only liturgical language, so local differences between the syncretic Manichean faith might become more pronounced over time.
- Conversely, one might imagine Aramaic becoming the liturgical language of Christianity, with it spreading to Persia, India, and Central Asia in the East, and North/Sub-Saharan Africa in the West. This might lead to a great deal of cultural exchange with the European Manicheans, considering Mani himself wrote in Aramaic, and Jesus spoke it.
- I wonder how the crucifix would be seen by the European Manicheans? By the 3rd century it had become a prominent symbol for the Christians (so you could imagine the Persians adopting it when they convert), but Jesus remains a prominent figure in Manicheanism, and Mani was said to have died on the cross as well. Depending on what symbols the Roman Manicheans adopt, there might be some cultural struggles over who has 'ownership' of the cross ITTL.