Since then, the two children lived alone.
From then on, the two children lived alone.
The first sentence isn't correct grammatically, but the second one is correct.
The idiom "from then on" means "from that time"; it's used for an action or event that happened in the past; the event or action is no longer continuing.
From then on, the two children lived alone. (They did so in the past; they don't live alone in the present).
On the other hand, the phrase since then also means from that time, but we don't use it in the past simple. It's used for an event or action that's continuing. Hence, it's used in the present perfect or present perfect continuous such as:
Since then, the two children have lived alone/have been living alone. (They still live alone).
(When I looked up the word "then" in the The Free Dictionary and Webster, I was surprised to know it's also used as a noun that means "that time" (a point in time for which we use since in perfect sentences).