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