Until recently, Apple only allowed users to use the Apple Watch as a mere remote for streaming Spotify via their smartphones. Apple just rolled out a server-side update to its watchOS app, making it extremely easy for Spotify users. Several users received a message from Apple Watches notifying them it was ready to stream directly from Spotify.
The Apple Watch requires one to have internet access for it to stream live, whether it be LTE or Wi-Fi.
In order to stream Spotify, users will come across a menu item the bottom right corner of the UI allowing them to interchange among the devices that actively have the app running.
Some users have complained about issues of the update not showing up. Although there is no official announcement from Spotify yet, word has it that the update should be available for everyone any day now.