Spotify updates your Discover Weekly playlist every week, on Monday.
Every Monday, Spotify resets the playlist and updates it with new songs. Once this happens, you won't be able to see the songs from your old Discover Weekly playlist. There is no way around this using the Spotify app.
Unfortunately, there is nothing you can do to get it back. What you can do, is sign up for Rediscover. It automatically saves a copy of your past weeks Discover Weekly so you have a history of all your old Discover Weekly playlists.
Unfortunately, the past playlist's songs are gone on Monday when Spotify adds the new weeks songs to the playlist. You can sign up for Rediscover to automatically have your Discover Weekly playlist saved every week. You then have an archive that has the full history of your old Discover Weekly playlists.
Rediscover uses the Spotify API to automatically make a copy of your Discover Weekly playlist every Friday. You can then see your old playlist for each week, even after Spotify has updated your Discover Weekly playlist.
You will be asked to login to Spotify and then grant permissions to Rediscover. Once Rediscover has the needed permissions, it works in the background to backup your playlists. You don't have to do anything else.
Rediscover will save the copy alongside your Discover Weekly playlist. It will be titled "Rediscover - January 1st", using the date that the playlist was created.
Yes, Rediscover is completely free to use. It is built, maintained and paid for by @rileytomasek.
Go to your Spotify account settings, find the apps tab and then "revoke access" from Rediscover. You will now no longer get new Rediscover playlists
Message @rileytomasek on Twitter and I'll help you.