Undead come in two varieties: sentient and mindless. This reflects the philosophical doctrine of mind/body dualism, and suggests that many undead may be incomplete beings. A ghost, for example, is a spirit or mind without a body. A zombie is brute flesh with animating will but no mind or soul.
What happens to the spirit when the body is called back to service in the case of the mindless undead? Does the campaign world have an afterlife or an underworld where the spirit resides? Is the spirit gladdened by the body’s return to service? What if every time a skeleton or zombie is animated, the ghost is also pulled back to the mortal realm, with no chance for eternal rest until the zombie is destroyed? Ghosts could be great allies to player characters.
Perhaps necromancy requires that the ghost be imprisoned, preventing it from reuniting with the body, resulting in necromancers collecting such trapped spirits (which would also be of interest to demons or other entities that might value such things). This would result in necromancers making many ghostly enemies, and might contribute to their terror of death and their own search for immortality and lichdom. Failing that, they would seek to escape the grasp of the underworld by true extinction of the soul. A sort of necromantic nirvana.
These trapped spirits could also be bound to objects. Maybe some or all magic items are the byproduct of such necromancy. FrDave had a post about a magic sword that was infused with the spirit of an elf-maid, and in D&D magic swords are traditionally sentient; this is another way to explain where magic swords get their sentience.
The most terrible form of zombie creation: tearing the soul out of a still living person, leaving only an undying and mindless slave.