How does Faulkner's "Dry September" highlight differences between the old and new South?

William Faulkner once remarked, "I love the South; I hate the South." This Nobel Prize winner who rarely strayed far from his home of Oxford, Mississippi, depicted the South with honesty and candor, but always there is a certain poignancy to his tales. In his works, Faulkner documents the ability to endure and illuminates social issues with honesty, sparing no level of Southern society. The Old South that has become decadent is depicted in the Compsons of The Sound and the Fury; the poor whites, "buckra," are portrayed in As I Lay Dying. Yet there is in both these novels a discussion of the existential metaphysics of everyday life. 

In his short story "Dry September," Faulkner unapologetically portrays the "good ol'boy" of his beloved South, a type character of the white male who dominated by any means necessary. The narrative revolves around making the innocent Willie Mayes the scapegoat for the questionable behavior of one of the white ladies of the town. Whether Mayes is innocent or not is ignored in the overriding passion to make an example of any black male so that others will not be tempted to cross racial lines. (Miscegenation was against the law.) When the barber tries to convince the old soldier McLendon that Willie Mayes would never commit rape--"I know Willie Mayes"--and that the incident probably never happened because Miss Minnie has "a bit of an imagination," he is contradicted:

Happen? What the hell difference does it make? Are you going to let the black sons---- get away with it until one really does it?

As the barber continues to try to reason with the men who have now taken on the mentality of the vigilante, he is told,

"Sure, sure," the soldier said. "We're just going to talk to him a little; that's all." 
"Talk hell!" Butch said. "When we're through with the..." 
"Shut up, for God's sake!" the soldier said. "Do you want everybody in town...."

They load into cars and speed off to grab Willie Mayes, who is a night watchman at a plant. Willie pleads with them that he is innocent, but to no avail. When McLendon returns home, he flings his wife brutally out of the way, takes off his shirt and wipes his sweating body after laying his pistol on the bed. There is no authorial comment until the last subtle line, "The dark world seemed to lie stricken beneath the cold moon and the lidless stars." 

The South of the early twentieth century was very cruel to African-Americans (it is noted that McLendon served in World War I). It was ruled by Jim Crow Laws after wealthy investors of the North, eager for the South to recover its economy as profits could be gleaned from the shipping of tobacco and cotton, helped to pass laws to undo many of the post-Civil War rights afforded African-Americans because Southerners argued that they needed to control this large population. Unfortunately, the acts of McClendon and the others who inflict such deadly cruelty upon a man are exemplary of means used to subject and control others.

But after 1950 and the Little Rock Nine, schools in the South became integrated as did other facilities. The turbulent 60s brought about the most important transformation, however, and it was in this decade that Faulkner died. After the legislation of the Civil Rights Act of 1964, the South was forced to change, and there is nothing uncommon about couples of differing races or children of mixed race in Southern towns.

How is "Dry September" a devastating critique of the South?

The main purpose of the short story “Dry September” is an indictment of Southern culture, specially racism.  In the story, a young black man is murdered.  He is accused of attacking an elderly white woman, Miss Minnie, but is completely innocent.

A common theme in stories of racism in the American South, when a white woman accuses a black man of anything he is innocent until proven guilty.  The story is an indictment of Southern culture because of the following:

1. A white woman accuses a black man of a horrible crime just to get attention.  Miss Minnie is an old spinster just reaching for the spotlight.  She basks in the glory after her accusation and the murder, then returns to being unimportant.
2. Black men are assumed guilty.  Will Mays is murdered for attacking a white woman even though there is no proof that he did it.  He is guilty, and no one bothers to see if he is innocent.
3. Vigilante justice in the form of white mobs killing black men.  There is no trial.  A group of white men kill Mays as soon as he is accused.
4. Murder of a black man is commonly accepted.  No one in the community seems to care that the man might be innocent.  Although they begin to suspect nothing really happened, they are not disturbed by it and Miss Minnie is just upset she is no longer the center of attention.

As you can see, all of these things are terribly wrong and are only present in a terribly racist society.