The idea of a demilich is kinda stupid to begin with, honestly. If you're such a powerful lich, surely you'd have access to the possession spell; just make a sick new body. or steal one, if you're too lazy to get into necrografting even with an unlimited unlifespan.

IIRC there are two kinds of demilich. One is the remains of a lich which has become powerful enough to transcend the need for a physical body, and is more powerful than a lich; the other is the remains of a lich who failed to feed their phylactery, and is less powerful.

Edit: 2014 MM page 49:

The immortality granted to a lich lasts only as long as it feeds mortal souls to its phylactery. If it falters or fails in that task, its bones turn to dust until only its skull remains. This "demilich" contains only a fragment of the lich's malevolent life force- just enough so that if it is disturbed, these remains rise into the air and assume a wraithlike form. The skull then emits a terrifying howl that can slay the weak-hearted and leave others trembling with fear. Left alone, it sinks back down and returns to the empty peace of its existence.

Few liches seek to become demiliches, for it means an end to the existence they hoped to preserve by becoming undead. However, time can erode the lich's reason and memory, causing it to retreat into its ancient tomb and forget to feed on souls. The spells it once knew fade from its mind, and it no longer channels the arcane energy it wielded as a lich. However, even as a mere skull it remains a deadly and vexing enemy.

And the sidebar:

The transformation into a demilich isn't a bitter end for all liches that experience it. Made as a conscious choice, the path of the demilich becomes the next step in a dark evolution. The lich Acererak-a powerful wizard and demonologist and the infamous master of the Tomb of Horrors-anticipated his own transformation, preparing for it by setting enchanted gemstones into his skull's eye sockets and teeth. Each of these soul gems possessed the power to capture the souls on which his phylactery would feed.

Acererak abandoned his physical body, accepting that it would molder and dissolve to dust while he traveled the planes as a disembodied consciousness. lfthe skull that was his last physical remains was ever disturbed, its gems would claim the souls of the insolent intruders to his tomb, magically transferring them to his phylactery.

Liches who follow Acererak's path believe that by becoming free of their bodies, they can continue their quest for power beyond the mortal world. As their patron did, they secure their remains within well-guarded vaults, using sou l gems to maintain their phylacteries and destroy the adventurers who disturb their lairs.

Demi doesn't exactly mean lesser demi means half or partially.

It's the last part of a normal Lich. Think of it like it's concentrating it's power.