50% reduction in build time for shrines and temples.
For early game, this is best used for getting shrines built in your cities. Temples are not needed early game to get a religion. Shrines in a couple of cities should be more than enough. Why would you build a temple that adds +2 maintenance when you are already starved for gold? The faster build times on temples is best used later, and is even better if you pick a belief that enhances the template (i.e. +2 happiness or culture to the temple). Faster temple build times means you get those faster.
If you're not going to build temples, then the benefit of 3 half-cost shrines in 3 cities is 60 hammers, which is literally the same as spending an entire policy to chopdown three sets of trees. That's
if you manage to get four cities early, a task Piety gives almost no help with at all. (It doesn't count in your capital since we've established that outside of a very early culture hut, you'll already have built your capital's shrine).
+1 Faith per temple and shrine
This is a huge boost to faith production. A couple of cities with shrines almost ensure you get an early religion, and enhance much faster. All the added faith also means you can buy faith-bought buildings earlier (giving you their benefits earlier), buy missionaries faster (to spread your religion earlier), and help accumulate a larger faith mountain for late game. Piety is best used with wide empires, and this extra faith generation will add up.
We've just established that you're not building early temples because you can't afford them. So for an entire policy point, you are getting, at best, an extra +4 faith per turn.
In contrast, a player with Dance of the Aurora and whatever that desert faith power is called is earning +1 Faith
per tile, cost free, with no build time except for needing population to work the tile (favor: Tradition). The Celts are getting +1 or +2 per city from their forests. Ethiopia is getting faith from steles--placed in cities for free (favor: Tradition, which grants them completely for free). It quickly becomes evident that Piety is not at all suited for a serious religious game. That honor goes to combining Tradition (maybe Liberty) with all the various non-Piety ways to earn faith.
Ok, so, in some games you don't start with an easy desert or tundra is hard to find, and you're not next to a good natural wonder. But the fact remains that with 8 civs in a game, someone likely will. And that player has been handed the most of the benefits of the Piety tree except for the Reformation Belief without having to go down it (in fact having several good reasons
not to go down it). Add to this the fact that Piety is all about the race in a way that neither Tradition or Liberty are--go down it and fail to get religion and you are screwed. So it's not even worth trying unless you are sure you will win something worthwhile. Because the fact remains that even if you just went Tradition and ignored Religion completely, you'd
still end up with bonuses because unlike other mechanics most of the benefits of Religion are transferrable.
As for buying units, I already explained why this is a losing deal early on. You need those Faith points for your prophets to justify having taken Piety. You
might be able to get to Hagia Sophia to get the prophet there, but Tradition would get you faster and give you a better chance of actually getting to build it. The discount policy provides a 20% discount on missionaries and buildings. Most of these at standard speed cost 200 Faith, the same cost as a Great Prophet. So you need to purchase 5 missionaries before you're caught up to where Tradition would have got you from the free prophet from Sophia. Add to this that by the time you actually complete your Policy tree, Tradition is likely an entire policy ahead of you. You are playing catch up with the Tradition player the whole time.
I'm not in the camp that thinks Piety is so awful that you should
never use it, but as an early game tree it is 99% awful. The sheer number of abilities that do nothing for you when you first take them stands at approximately 4 out of 6.