I’m not quite sure why Moments looks so much different as an embed on a website than they do in the mobile app. For one, it’s not a consistent look across devices, and doesn’t motivate publishers to embed them since it doesn’t fit within a reasonable amount of containment.
If Moments embedded on the web looked more like the mobile in-app format, it would be much easier to consume, but also provides ad inventory which can be split with the publisher. See this article - https://austinstartups.com/monetizing-twitter-1f29dfe51fb6