KaiserJohan speaks the truth.
As an additional note, for Catholics it also helps out with papal influence, or in the case of you being at odds with the Papal States (by say, attacking him for land to keep Italy in the empire) it will protect you from being excommunicated. (The Defender of the Faith cannot be excommunicated by the papal controller, though sadly claiming the title defender of the faith doesn't clear an existing excommunication so you'd need to plan ahead and take it before getting the negative relations with papal states). Of course, if you're looking at having negative relations with the pope for extended periods of time you're probably better off not being catholic when that becomes an option later on. Otherwise, I find the +1 papal influence and +1 prestige to also be helpful bonuses, though somewhat less effective than the Monthly War Exhaustion reduction and such.
Also, yes, there is an achivement for becoming the Defender of the Faith. The only requirement is to click the button to claim the title of Defender of the Faith while playing on achievement compatible Ironman and you will have gained the achievement "Defender of the Faith".