When you cancel a music subscription, Apple does not save any of your information and the playlists and downloaded music will be removed. There have been complaints that other companies are saving your information even after you quit using their service, and that has been seen as a privacy issue.
If you don't want that to happen, don't let your subscription lapse. If you think Apple should be saving your information even after you are not a current subscriber, then you are probably making the right decision to not use the service. You can let Apple know how you feel here:
Product Feedback - Apple
You can request a refund here, but that is not guaranteed.It appears you used the service just fine the last month and had complete access. You only own the music that you purchase and music downloaded from any streaming service is not yours, so there is no money lost there.
Request a refund for apps or content that you bought from Apple - Apple Support