The native client has application level access to the rest of your machine. They use this to run process loggers "for the activity display", or the button that allows you to quickly stream a game if it's running. They could theoretically use this access for keylogging or accessing the mic without explicit user permission. Running the Discord web client keeps the source of collected telemetry within the webbrowser, which doesn't offer keylogging or process logger features, and requires explicit user permission to give websites access to a microphone, camera, or the screen for streaming.

Yes, they do process log on the native client, and from my own GDPR data request it appears they keep this data in detail for a couple of years: https://github.com/snapcrafters/discord/issues/43

1. better privacy as no process scanning or direct access to cam/mic
2. better performance as discord desktop app for windows still uses 32bit electron, which makes it slower than the web app

In Linux the native client is quite bad,especially streaming, as its not hardware accelerated and doesn't stream sound. The browser version doesn't have any of those issues.