Long press on the Spotify icon, tap the (i) in the pop-up. Tap Notifications. First, confirm that all the boxes against the categories are checked. Next, tap on the text "Playback" (not the checkbox). You will be taken to a new screen for customizing notifications for this particular category. Tap Advanced, and see what 'On lock screen' is set to. Is it set to 'Don't show notifications at all'? If yes, change it to 'Hide sensitive content', and you should see playback controls on lock screen too.