The trade-off is a bit of CPU time, every time the page is referenced. Not just when it is updated or read from storage, but potentially every time it is accessed, even just reading, as the pages held in RAM are in the same format as on disk so the decompression needs to happen on read also (unless postgres keeps a cache of pre-decompressed pages to help with this but my gut suggests that would not be worth the complication - where it would help significantly you would just not use this technique anyway).
In a great many scenarios, I would say most, this trade-off is worth it because of the relative cost of IO and CPU time needed to save some of it.
It might be significantly detrimental to performance in situations where enough of your data fits into RAM anyway and you are performing CPU-heavy analysis upon it.
Side benefits include reducing the size of your backups and reducing the time needed to take them, so there are other potential infrastructure and administration savings to be had too.
The index also has to be read from disk (if not already in memory), which is faster the smaller it is. But also, once in memory, smaller size makes key-lookups or traversals faster.
How much faster? It really depends on your index.
So with Postgres I try to keep the number of indexes per table down to around 5, but with MySQL InnoDB, 20 indexes is not a problem for many use cases.
The other little-known Postgres problem is that by default, installing it sets up replication that writes a local file in a busy manner. This can cause problems with fsck or on low-end SSD/flash storage resulting in corruption that causes postgres restarts to fail hard, so disable that.
And if you're getting corruption in your database on low-end storage, that storage is simply broken. The database can't do anything if your storage system lies to it, or writes different data than what you give it. It is possible to configure Postgres in a way that leads to data corruption when e.g. the power goes out, but that is an option with a big warning that you should never use outside some specific circumstances. The Postgres defaults don't corrupt your database if your hardware works correctly (absent some pretty rare bugs in Postgres itself).
Why Uber Engineering Switched from Postgres to MySQL (2016): https://news.ycombinator.com/item?id=17280239
A PostgreSQL response to Uber [pdf]: https://news.ycombinator.com/item?id=14222721
One important point is that Uber wrote this running on Postgres 9.2. The latest Postgres release is now 12.4
Just a minor nitpick: Postgres changed the version numbering scheme with version 10. So 9.4 was a major release, but 12.4 marks a minor (bugfix) release.
If you want to name minor releases it would be e.g. 9.4.26 vs. 12.4. If you want to compare major versions it would be 9.4 vs. 12
9.2 (released in 2012)
9.3
9.4
9.5
9.6
10
11
12
13 (about to be released this year?)
That's still a lot of versions in between, each of them as new functionality and requires converting the database to upgrade.
I'd say it was a nice example what happens when you design application based on how you imagine an ideal database would work, not realizing that there's a lot of complexity and limitations. And then when you eventually have to face reality you blame the database for having to do trade-offs and refuse to mitigate the trouble by changing the application.
Another time (I think it was a MariaDB) the database get corrupted every few days. First we though it was hardware, so we replaced disks, then the entire machine. But it continued to happen. Eventually it turned out it was a bug and with specific pattern written to the disk, when MariaDB read that block back it incorrectly thought the block was encrypted and discarded it. Updating to a version that fixed it, responded the issue, but that's really a nasty bug that caused data corruption.
Postgresql on the other hand with my experience always was nice even in bad situations. When we had network issue (due to failed switch) one the problem was fixed the database reconnected back and trained replication, MySQL in similar situation required to tell it to reconnect again, some even setting up replication again (which requires restoring backup on slave and reconfiguring replication again, which required few hours).
In another instance we had main postgresql set up replication to a DR site. The database data was stored on filesystem provided by NetApp. A consultant had a task to deprecate filesystems that were no longer used, so he actually put them offline for 30 days, before deleting them. Around 20 days in we noticed that the master was running out of space. Turns out the disk was utilized by unshipped logs to the standby. Once filesystem at the DR site was put back online everything resolved all by itself.
Firstly, it's not quite true "Uber migrated away because of write amplification" because they named multiple other reasons contributing to that decision. (And yes, not all of those reasons were quite valid, but that's a different discussion.)
Secondly, it's patently false that Postgres sets up local replication writing into a local file. That's plain bogus.