Is this actually an extra layer of security? Is it harder to get read access to another user's file with 0600 permissions, or to read memory from another user's process? In my limited understanding they seem equivalent (at least on Linux).

They're approximately equivalent on Linux. Adding a password helps if you accidentally tweet or email your private key, leave it on a flash drive, leave your computer's disk decrypted, etc.

It's primarily targeting offline attacks ie: I've stolen your laptop or I am accessing it when you haven't decrypted it.

But there are also niche situations where an attacker has read permissions for files but not memory. It's atypical, but in cases where you have 'confused deputies' it happens - for example, if I have a service that attempts to safely broker reads to files but that service has a logic bug where a client can trick it into reading sensitive files.

The classic example of this is a path traversal in a web service.

Also, while there's no boundary, attackers are a lot happier reading files than memory. They're only human.

In general I find that, when you can, you should just encrypt your data. It can protect you in some surprising ways.

Encrypting the file, or better whole disk encryption, prevents someone stealing the laptop and yanking the drive to get access to everything.