Software running in a browser is probably going to heavily sandboxed and have tight restrictions on what it can access since the purpose of a browser is to run untrusted code. Native software will usually have minimal sandboxing because it's supposed to be trusted, and can do anything the user running it could do.

As far as performance, I think that only applies to apps like Slack and Spotify that use Electron to pretend to be a native app.

Which do you think is more potentially damaging?

1) Using a website which has had its server code compromised (slack).

2) Installing and using an application which has had its code compromised (maybe also slack).

The installed application is going to have more access and potential to damage your system and to compromise your data. There's not really anything more to it. One's in a browser sandbox and limited by browser capability, the other can do literally anything it wants.

What's the mystery here? It doesn't have access to your file system or the ability to request root access.

I expect he's referring to the web page running in a sandbox vs running an app locally with no sandbox.