A deteriorating patient is physically trapped by a limb where they will almost certainly die during the time to secure extrication
A completely mutilated non survivable limb retaining minimal attachment which is delaying extrication and evacuation from the scene
This paper provides an overview:
Porter KM. Prehospital amputation. Emerg Med J. 2010 Nov 13;27(12):940–2
and is summarised in this blog post.
These videos demonstrate the technique: