The two of them generally mean the same thing, and can quite correctly be used interchangeably. So:
Since then, the two children lived alone...
Means the exact same as:
From then on, the two children lived alone...
But, there is a very slight difference here. Since then can mean that the action happened at some point in time after the event. I.E, at some point after their parents died, they started living alone. From then on does not give that kind of meaning that from the very moment, they lived alone.
However, there is a possibility, that in certain situations, from then on can be taken to mean from that moment until another, when used along with other words. For example:
From then on until the day they (did something), the two children live alone...
This sentence style isn't often used when using since then, because since then is often understood as from one moment until the present. However this is ambiguous, since one can still use the word until to give the same sense.
Since then, until the day they (did something), the two children live alone...
In conclusion, if you meant that they lived alone from the very moment their parents died, from then on is slightly more correct.