While it's working correctly (which involves you knowing the private key), you know everything it sends. If it gets compromised, it might keep using the same key and you know what it sent, or it might start using a different key and you'll never figure out what it sent while compromised.

When a server does use forward secrecy, it looks like this:

While it's working correctly (which includes logging session keys), you know everything it sends. If it gets compromised, it might keep logging session keys and you know what it sent, or it might start logging fake keys and you'll never figure out what it sent while compromised.

What meaningful difference is there?

You don't trust the server to tell you what it sent. You record what it sent, and at some point you can check if it decrypts or not. This acts the same whether or not you have FS enabled.

> The objective is to be able to, after the fact, prove that you could trust it at that point in time.

You can verify what it sent, but that doesn't prove it wasn't compromised.