The Sandman is finally a TV series in a version with Neil Gaiman's direct involvement and a faithful adaptation, but it was never going to be 100% faithful. Changes were always going to be made to fit the story into a television series format, with parts of the original comic that haven't aged well being brought up to date. From this point forward, consider the "MAJOR SPOILERS AHEAD!" sign on and we'll meet you on the other side of this spoiler image buffer…
We're not interested in talking about casting choices, but story changes to lend the series more dramatic heft or thematic cohesion. Where The Sandman comic had a looser narrative and played with randomness, the TV show edits, revises, & compresses many plot elements while retaining the ideas and spirit of the story.
In The Sandman comic, Dream was returning from a battle in a weakened state that made him fall into Burgess' summoning trap. In the show, Dream (Tom Sturridge) comes to the waking world to capture the Corinthian (Boyd Holbrook) for running amok and killing people in the waking world and gets trapped by Burgess's spell. This was taken from the start of the 2014 comic miniseries The Sandman: Overture, where Dream summons the Corinthian in 1915 with the intent to uncreate him but is called away to another crisis, the battle that forms the backstory of the first issue of the original comic. In the show, Dream is trapped by Burgess' spell before he can deal the killing blow.
The Sandman television series makes the Corinthian the Big Bad of the season. It gives him motivation: to stay in the waking world and pursue his pleasures, so he does all he can to keep Dream imprisoned. He helps Roderick Burgess keep Dream prisoner. He later helps John Dee (David Thewlis) escape to possess Dream's ruby. In the comic, Dream getting held prisoner is what enables the Corinthian to run wild without fear of punishment. In the show, the Corinthian takes an active role in manipulating events to hinder Dream.
In the early issues of the comic, Dream travels alone. The pilot episode of The Sandman shows Dream's previous raven companion Jessamy's death when she tries to rescue him from Burgess' prison. Sturridge's acting indicates Dream had an emotional attachment to Jessamy, so her loss makes him reluctant to accept Matthew (Patton Oswalt) in Episode 3 as her replacement.
In the show, Ethel Cripps (Niamh Walsh) steals Dream's tools and runs away from Burgess' cult on her own rather than with her lover. The show doesn't reveal how she ends up with the demon's amulet that protects her, though it was probably scripted and filmed but cut for time.
Roderick Burgess is an Aleister Crowley manqué and rival, and in the comic is depicted as a more craven comic book villain, murdering his rivals and enemies with magic. The show gives him more stature and is less cartoonish, and Charles Dance gives him some weight. His grief over his dead son is more upfront than in the comic. His surviving son Alex (Laurie Kynaston), is less the power-hungry leader than a fearful son trapped in his father's shadow and the cycle of abuse.
Roderick Burgess' death is more directly at the hands of his son Alex, albeit accidentally, than just old age and a heart attack. The story builds up Alex Burgess as a more nuanced character, a man whose moral cowardice defines his life. He killed Jessamyn because he was too frightened of his father. He didn't free Dream because he was too afraid of Dream's revenge. In the comic, Dream punishes the elderly Alex for guilt by association and moral complacency. In the show, Dream punishes him for a lifetime of moral cowardice and for punishment for killing his raven.
In the show, Dream's escape from his prison is largely faithful to the comic, though only two guards are present when he breaks out.
"Sleep of the Just" looks and feels like a live-action version of the original first issue of the comic, but it can only go so far in replicating the claustrophobic, distorted expressionism of Sam Kieth and Mike Dringenberg's art, which recalls Bernie Wrightson's horror comics from the 1970s. It works hard to capture as much of the book as possible, though, and it's going to be interesting to see what further changes will see on our screens.
The Sandman is now streaming on Netflix.