The difference is GKV being the default, and opting out of that kind of hard. You have to earn €5,062+ per month, for a start [0]. And the step is intentionally hard to reverse. I consider that a decent compromise. At that point people have to actively shoot them self in the foot, and I've little pity for those.
If you didn't opt out, thus are still under the GKV, the system will cover emergency services. Even if you haven't payed (yet).
[0] https://www.bundesaerztekammer.de/weitere-sprachen/english/h...