I like to differentiate the different spiritual undead by imagining that there's a whole unseen "ecosystem" ("thanatosystem?") going on in the spiritual world, and the spirits are products of that.
Ghosts are a default, "natural," product of a sentient being dying. The process of dying while still emotionally tethered too strongly to the physical world either traps the living being's spirit in the ethereal plane, deprived of a physical body, or it actually creates a new incorporeal creature that is a copy of the original being's memories, personality and motivations at the time of death. Which is the truth is up to the game master, but may also be a complete mystery, even to the gods. If you follow a mystical paradigm like that of the ancient Egyptians, people may have multiple souls, and a ghost is just one of those souls, left behind when the rest have departed. The end effect is that a ghost is, by and large, the same being as the creature who died, mentally speaking. It acts the same, and probably thinks that it is the creature, whether it is or not. Of course, being turned into an incorporeal entity of pure emotion and ether is pretty traumatic, and most likely leaves very few of them sane.
What is a specter, then? Since there are evil, vengeful ghosts, a specter must be something distinct somehow. I like to think of it as a kind of parasite. When a ghost is created, and at it is full of rage and hatred, probably due to a very horrific and traumatic death, perhaps that vitriolic emotion draws beings that roam beyond the ethereal plane and can normally never interact with the world at all. They latch onto such a ghost and feed on it, destroying all remnants of positive emotions and personality traits, transforming it into a dark parody of what it could have been as a ghost. And, this unnamed parasite that infests it is native to, or connected to, the negative energy plane (or whatever your necrotic cosmological equivalent is), it carries with it a desperate need to feed on the energy of the living to fill its forever empty, hollow core. This type of spirit is a rabid wolf of the ethereal world, a danger to all that lives. Using this model, over time, normal ghosts may be in danger of becoming specters if they indulge their darker emotions too often. Specters can reproduce in a fashion, ripping the ghosts from the living beings they kill and infecting them with their dark parasites. This may be the only way for these parasites to reproduce, making the living world their spawning ground.
Wraiths are even less human than specters. Probably the only link they have to the living being that produced them is a vague, twisted visual resemblance; nothing of the individual's personality remains. There is only a horrible, alien, unquenchable thirst for life energy and enough intelligence and cunning to hunt living beings in a (futile) effort to sate that inner hunger. This makes them similar to specters, and therefore are likely victims of a similar negative-energy (necrotic) parasite. But unlike the parasite that produces the specter, the seed of a wraith is far more focused and predatory. It doesn't just infest ghosts, it transforms them into its image. Fortunately, this makes them more limited in many ways, lacking the personality and adaptability of the more self-aware specter. They reproduce in a similar fashion to specters, but are much less likely to spontaneously infect ghosts, instead most often coming into being when a wraith kills a mortal, or they are created by necromantic magic. Where a specter is a rabid wolf, a wraith is a shark, something that doesn't seem to have any nature other than hunting and killing.
I use "origin stories" like these to form the basis for descriptions of these creatures. Ghosts are, as always, human-like. Specters seem like ghosts seen through a dark, cracked mirror, filled with and surrounded by hungry shadows. Wraiths are like incorporeal predator animals, human-shaped voids of shadow highlighted by glittering teeth and coldly glowing eyes.