Getting songs to download/sync on your iPhone is a constantly recurring problem with Spotify.  Nothing drives users more crazy than hitting download on a playlist only to find the songs aren't downloading, meaning they aren't playable on your iPhone.  And the "solutions" to fix this problem that are offered on this forum range from the laughable (turn your computer off/on - delete Spotify and reinstall) to the downright dangerous (I've seen people recommending your mess around with or delete files on your computer's operating system.  Crazy and irresponsible nonsense!). 

Here is how you ACTUALLY solve this problem and it is SO SIMPLE.  

Go to SETTINGS on the Spotify app on your iPhone.  Scroll down to LOCAL FILES and when you open it make sure the toggle is switched to "enable sync from desktop."  (This enables discovery of local files to sync from a desktop machine on your local network running Spotify).

If this isn't turned on - you CAN'T DOWNLOAD/SYNC songs - it simply won't work.  Yes, it's that simple.