I usually use Missionaries to spread my religion.
For most of the game, Missionaries are cheaper than Prophets. I like to spread my religion early, so Missionaries are a better bet. Missionaries are faster (4 movement compared to 2), which means they reach destinations quicker, and can maneuver around other units more easily. As civilian units, neither Missionaries or Prophets can move through other players' units and I've occasionally found my Prophets stuck due to other units. Also, Missionaries are more disposable so if they get captured by barbarians or enemy civs, it's less of a loss.
Great Prophets are expensive in the early game. I like to settle my Prophets as holy sites. Prophets don't get weaker with attrition (which can kill Missionaries) so they can penetrate deeper into enemy territory (relevant if you don't have Open Borders). They do a much better job converting cities than Missionaries because they also remove the followers of the other religion. It's not necessarily an auto-convert but usually is and can come close in other cases (recently I spread religion to a city state with a population of 17 and a different religion. It ended up with all of the opposing religion removed and 8 people of my religion, so close to conversion but not quite).
Missionaries have a fixed cost for a given Age, so you can spam them out. Each Great Prophet you generate increases the cost for the next one, so it becomes progressively more difficult to churn them out.
In the early game, when cities are small and other religions are not present, Missionaries are the most efficient. As eras progress, Missionaries become less cost effective and you might be better off saving your faith for Great People. Also, as cities get bigger and especially when other religions are present, it might take Missionaries many attempts to convert a city. At this point, Prophets become better because they can usually get the job done with one try.