The Ottomans were already pretty heavily Romanized. In this scenario, they're even more a continuation of the Roman Empire. There's less Persian influence and much less Arab influence on the resulting empire. The world loses a chunk of Persian art and poetry. Russia is more heavily influenced by the Middle East in TTL, since once the Golden Horde go down the Christian Ottomans will pull Muscovy into its cultural orbit. I wonder if these Ottomans end up pulled more northwards... Iraq was a poor province throughout the Ottoman period and since it's full of Muslims, it's easy to see a Christian Ottoman Empire deciding it's too much trouble and leaving it to the Persians... And instead we might have an Ottoman Ukraine and Volga delta.
If the Ottomans inherit the Byzantine-era attitudes towards Copts, a conquest of Egypt could end up being unprofitable. We might see things looking alot like OTL's Ottoman Egypt as the Muslim Mamlukes retain their position due to not being Copts... If the Ottomans are more religiously tolerant, then Egypt and Syria become more interesting - both areas had significant Christian but not Orthodox populations. If the Ottomans manage inter-Church relations well, then the mind boggles at the interesting directions opened up.
Bosnia may never become so Muslim in TTL. OTL Islam spread really well among the Bogomil heretics. If the Ottomans aren't Muslim, likely Islam doesn't spread and we again come back to if the Ottomans allow the Orthodox Church to control their policy towards heretics, or if they resist and take a more tolerant approach... Then again, maybe the Ottomans remain very pro-Islam in order to ease the transition to Roman-ness for their many Muslim subjects. In this scenario, I suspect that Ottoman Islam would be a more positive version of TTL's Ottoman-era Islam (religions generally are less corrupt when the power of the state is locked away from them), so more divergent views, less corruption in religious foundations, less theology students participating in coups. So we could see Christian heretic groups converting to Islam anyway.
This Ottoman Empire would be even more pulled towards involvement in Italy.
If there's a Safavid revolution in TTL, it's probably even more successful, since in TTL it will be a revolt against a Christian Roman Emperor, not a revolt against a Muslim Caliph. Of course, it's over 200 years after your PoD, so probably we see some different revolt of the tribes in the mountains of Anatolia, and likely, the revolt is earlier.
fasquardon