Basically, the download button is a single down-arrow at the top of your library. When you tap it—and download your music–the down-arrow turns solid green to indicate your music is available offline. It doesn't disappear. It turns green, Spotify's primary color.
The kicker is: if you accidentally (or out of curiosity) press that green arrow again, it will wipe all your downloads without confirmation (disabling "offline mode"). Did that on a flight once and lost all my music. It's absurd how that UX made it out of testing.
Here's a picture of what I'm talking about. Tapping this icon deletes your library from your device: