It’s about the same as the desktop app. It doesn’t feel like a web app. The desktop app is better because it doesn’t feel out of place.
It’s not a bad UI/UX IMO, but it can take some getting used to. For the notifications I have to check several options such as “silence @everyone and @here”. Sometimes I find the updates annoying. But it’s among the best chat UIs I’ve tried.
It's definitely good UX for a chat app. It was good UX from the beginning, when they had a dark mode before a light mode and made voice chat not require installing anything (which was many peoples' first real-world exposure to WebRTC technology).
It's poor UX for a support forum where you want communication to be one-to-many as much as possible (to spread info to the largest number of people with the smallest amount of info producer effort as possible).
Those actually are bad UX, imo. In the servers I've seen implement them, they devolve into basically chat anyway. People still re-ask questions because the search functionality is the same as searching chat, which people weren't doing before. The layout still favors short messages over in-depth posts.
(I do find Discourse to be among the worst web forum software, and I can see similarities between Discourse and the Discord forum-style channels, so some of this may come down to personal preference. Discourse always felt too recency-biased and ephemeral to me compared to e.g. MyBB or phpBB, for example.)
Keep in mind also that it's still behind the walled garden of Discord's authentication and hosting. Imagine if you had to install the Tapatalk app to actually access any InvisionFree/ZetaBoards/Tapatalk/etc forums back in the day. It was annoying enough that mobile users were nagged to, but it would've been unthinkable to require desktop users to. (And being able to launch the app in your browser, while removing friction to "installation," isn't the same thing as actually being on the web-- search engines can't index Discord channels, Discord messages don't have human-readable URLs to share elsewhere, etc.)
Edit: Also, very relevantly to this thread, you can't just grab a webpage snapshot to archive a Discord channel like you could with forum threads. You've got to either take a screenshot (while dealing with scrolling) or scrape the data via the API, being careful not to trip bot protections or violate ToS, and then figure out how to present it separately.