As nouns, pall and pallor are unrelated. Pall refers primarily to a cloth draped over a coffin, and this definition gives rise to metaphorical senses: (1) something that shrouds or spreads over (e.g., a pall of fog), and (2) a gloomy atmosphere (as in the phrase cast a pall over).
Pallor refers to a pale complexion, especially one made pale by sickness, fear, or worry. And the word is often extended to refer more generally to the shade of the face, even if the shade isn’t pale.
Pall and pallor are distantly related where pall is a verb with several old senses having to do with weakening, becoming pale, or becoming boring. The verb is etymologically unrelated to the noun pall—which descends from the Latin pallium, meaning cover—and instead belongs to the family of paleness-related words that also includes pale, pallid, and appall.
Examples
Pall refers to a coffin cover, something that covers or shrouds, or a sense of gloom—for example:
Flowers are never placed on top of the pall. It is important that the pall be clean and free of wrinkles for each funeral. [United Methodist Altars, Hoyt Hickman]
A pall of smoke now hangs above much of the state. [Telegraph]
After the announcement of the split, a pall hung over the paper. [Daily Beast]
A pallor is a pale complexion (or, more generally, a complexion of any shade)—for example:
When assessing for pallor in darkly pigmented clients, you might experience difficulty because the underlying red tones are absent. [Transcultural Concepts in Nursing Care, Margaret M. Andrews and Joyceen S. Boyle]
She is beautiful – though she does not know it; with the pallor of one who barely goes out by day; eyes blinking with electric stars. [Express]
And the verb pall means to weaken, to become pale, or to become boring-–for example:
When conversation palls the favorite gambit at Moscow cocktail parties is to say, “Excuse me. I have to go collect a few rumors.” [New York Times]
They could add that the joke of having a pundit who bellows at the camera and has mutton-chop whiskers … has palled after nearly three decades. [Independent]