It depends on what beliefs you select and your overall strategy.
The science bonus to spreading your religion might provide impetus to send out missionaries. Other bonuses are calculated on the # of followers in foreign cities, etc.
Yes, I suppose if you figured out an opponent was going for a cultural victory, you could spread a religion throughout their empire that provides 0 cultural bonuses. But surely, that's an extremely weak incentive at best, and only is effective if they've established their own cultural-centric religion. The amount of resources it requires to grow and spread religion is enormous and could more effectively be used to sabotage that opponent through other means.
I think that the main problem with spreading religion is the super-high cost of missionaries . If their cost is reduced by at least 50%,it'd be much more interesting to try playing a non-founder style . Right now,you'd have to literally pray for the ai to send missionaries to your lands if you want to skip religion .
I don't see the advantage. Anybody?
I will say it here and now, misionarys are total waste of faith period!! Just simply use GP's and time, once you get it up and going it will spread. You need to keep your main cities clean and it will flow out. Pain in the ass for sure, but if you have units around your whole city, no way they convert. Not optimal for sure, and not to be done with more then 4 cities, but it works. After you get printing press, the pressure of religon gets insane. Use your gps to your advantage, the 3rd one save him for new religions, go to that city convert and profit. You basicly need to stop pressure, and the best way is to slow down the holy city, I know this works, because Songai did it to me, that is how I learned the trick.
Jaqpike: the main disadvantage of having a competitor's religion in your city is twofold:
1-They get benefits. Lots of belief benefit from having a lot of follower, wich help your competitor.
2-You dont get to choose the beliefs. You do get some benefit from the religion that is spread trough your lands, but some of them might be totally useless(having bonus culture for each plantation wont really help you if you dont have any plantations.)
2b-One belief give bonus to any units fighting nearby a city with their religion, and yes, it does work if your competitor attack you!