I have a soft spot for Persia among other countries so I'll comment a little.
1. The population of Persia was so thoroughly decimated by Mongols and in the end Timur that they didn't reach the same level of population until late 19cent. That alone says much. ( And when you read about Ahmadinejad's wish that Iran reaches pop of 150 mil. one day you can see that left a scar on their psyche as "Tatar yoke" did on Russians ). So stronger, more populous state that would be able to wage wars against any Ottoman or Anatiolian/eastern Mediterranean based empire on equal terms, not from defensive perspective as for example Safavids did IRL.
2. More prominent assimilation of the peoples of "Greater Iranian" family since stability of the state is ensured. If they would manage to hold Iraq heavily persianised pop would sprung up ( centuries in teh future ) there as well, maybe also even everything till the Indus river.
3. Not so sure what would be their relations with Egypt based Ayyubids and remaining Crusader states/cities.
4. Also if they survived and became one big, fat, stable empire when Portugese came into Indian Ocean for the spice trade I don't know how would their relations be.
5. If they go Shi'a then the stated branch of Islam would be even stronger than it is.
This is all coming from the perspective of them surviving for centuries to come and stabilizing into long a lasting dynasty/state aparatus.