The difference with PayPal is that they manage the subscription, and you're saying "stop my subscription", credit/debit cards have no concept of a subscription, so when you stop paying someone with one there is no communication of that to the seller.

Example: if you stop your gym membership with PayPal, the gym's membership system will either be notified that your membership has been cancelled, or they will do some periodic check to see who has an active membership and update from that.

If you stop your payments going through on a credit/debit card, it's essentially the same as having no money in your account, or your card being stopped because it's stolen. There's no communication back down the channel, you just can't bill it. As a result your membership starts to go into the red, which is usually allowed for a period because of payment issues. When they contact you with a bill months later they have every right to claim that money because you never ended the membership and could have been using the gym during that time. It is usually only through wanting good customer satisfaction that they might waive the fees for that period.