Burn through produces a hole through the root face, the pipe wall, or thickness of the base metal. Imagine trying to weld thin base metal with the current set too high. Usually it is associated with excessive heat input when depositing the root bead or the second layer (hot pass).
It isn't the same as melt through, which is weld metal protruding past the root when depositing a weld from one side of the joint. Melt through is often associated with single sided groove welds.
The difference between melt through and burn through is melt through is not necessarily a bad thing. It simply means complete joint penetration has been achieved. Burn through on the other hand results in hole through the joint that usually has to be repaired to keep the joint fluid tight.
ASME B31.3 limits melt through, but refers to the condition as internal protrusion.
Best regards - Al