On Android devices browsers, clicking on an embbeded Spotify Play Button should open native Spotify app. However, most times this does not happen because the browser throws an error: 'Page not available'.
To test this please visit Spotify Play Button page on an android device: https://developer.spotify.com/technologies/widgets/spotify-play-button/ and press play on the sample embedded player.
The browser will try to open this URL: intent://open.spotify.com/
My experience is that the 'intent:' fails to open the app most of the times. It worked a couple of times!!
Tested on a Nexus 4, a Nexus 5 and a Motorola Moto G 2nd Ed. All runnig Android Lollipop 5.0.x and Chrome V40.0.x