Thanks for reaching out. Could you check if you have any other songs in your queue? You can check that by clicking the three-dot menu inside the now playing screen and clicking on Go to Queue. There, if you see songs that's not in the playlist you wish to listen, skip until they all go away. After the list is clear, Spotify should start playing songs from your playlist.
This usually happens if you queued songs from a complete album or a playlist, causing all of them to play until they're all over.