When Wrath discloses his nature to Mustang, he describes what happened after the incident. First, keep this quote of his in mind:
The Philosopher's Stone is created from the life force of countless human beings; it contains their souls.
Now, consider that Father was attempting to put all of his wrath within one being; this would require that he removed not only any wrath from within himself, but also every single wrathful soul that composed him. That means we now have a Philosopher's Stone made from dozens of wrathful souls. What does wrath do? It gets revenge.
Countless souls have battled for dominance within me. And only the most wrathful one survived.
This means that he had no healing powers because he basically had no souls to regenerate from; only one soul remains within his body (his own or someone else's), and it is not enough to both regenerate and sustain him. He is basically a human with homunculus-level skills.
I also agree with Madara's conclusion that Father would want the public figure to age and appear human, so he may have been perfectly okay with Wrath being composed of only one soul. Pride's disguise was already conspicuous enough; you can imagine the uprisings if the military leader was also accused of such things.
With regard to why he is different from Greeling, his soul was pushed aside from the infusion, whereas Ling's was strong enough to coexist with Greed's.