First, your skepticism is warranted, as at least one definition of each word uses the other word. From the Oxford English Dictionary for "privilege, n." and "prerogative, n.":
Privilege, 2.b. The special licence, prerogative, or immunity attaching to a specified office or rank. Also fig. and in extended use.
Prerogative, 2.b. gen. A special right or privilege possessed by any particular person, group, class, or institution.
So this could be an instance of tricolon (three items of equivalent length or structure listed) where two items are so similar to emphasize ecclesiastical privileges. Compare with The Gettysburg Address, delivered by Abraham Lincoln:
We cannot dedicate, we cannot consecrate, we cannot hallow this ground.
Dedication, consecration and hallowing all suggest treating the ground as holy or sacred through a rite, and so the tricolon has emphatic force through the repetition. It sounds like Trollope is creating a repetition here too, weaker in force, where the full perquisites and functions given to clergymen take up much of what the High Church does.
Otherwise the shades of difference in tone are small. Some prerogatives have a claim to being inherent or at least prior to others, like the royal prerogative in the British monarchy, defined in Merriam-Webster as
the discretionary power inhering in the British Crown.
The commenters associating prerogatives with authority or rights may be drawing from this sense.
Meanwhile, many privileges are granted, meaning that they come from some higher authority, as with this Merriam-Webster phrasing:
a right or immunity granted as a peculiar benefit, advantage, or favor.
Again, this is a small difference and not a definitive one, touching (at most) the connotations of each word.