tl;dr: This show is fantastic, both from the point of view of an interesting mini-series and as an adaptation of a complex and unique novel. I honestly had no confidence they could translate the dry, surreal British humor to the screen, but Gaiman did it. By his own admission, he wrote it in honor of Terry Pratchett, so that it would be something that Pratchett would be proud of, and kept as much of the original plot as he could. Sure, some things had to be cut, and new thing were added, but the story is tight and the spirit is spot-on.
Now for the longer review, with spoilers:
I think the thing for me that really grabbed me about this adaptation is how seamlessly new material was woven into the existing backbone of the book. Granted, I'm not an expert on the book, but perhaps it's a good thing that there are many things that I'm not sure if they were in the book or not, meaning that the flavor of the new things matched. Gabriel, for example, is a completely new character, the angel leading the force of heaven into Armageddon, designed as the most irritating boss anyone could ever have. I know Gabriel wasn't in the book - there were few named angels in the book except for the Metatron, if I recall correctly - but he certainly could have been. Best of all, he wasn't simply created as another humorous character in a show with an already enormous cast list - 240 characters, I've heard. His purpose was to fuel Aziraphale's indecision about what to do about the Antichrist and Armageddon, to make him question whether he should be an obedient angel and let the world end or join Crowley in attempting to preserve the world, and that indecision nearly doomed them all. (Oh, Aziraphale, if only you had been paying attention to "Hamlet"...)
The show also managed to preserve one of its qualities that endeared me to the book: its depiction of minor characters. With only six hours to tell the main story, it couldn't do it much, but it did flesh out a number of the minor characters in subtle ways. The most poignant, I think, was the international delivery man, giving a glimpse of his home life before he went out for his final two jobs, but even characters with a handful of lines, such as the man who delivers the package to Newt and Anathema at the end or the guard on duty at the airbase, are unique enough that you gain a good sense of who they are in just a couple of minutes.
I think the show's greatest achievement though, was developing Aziraphale's and Crowley's relationship over six thousand years while keeping the modern-day story of Armageddon moving along. Both angel and demon change subtly as they spend time on Earth away from their own kind, and though they meet up occasionally on civil terms, they aren't willing to admit that they might actually like each other, even while helping each other out and saving each other's, uh, corporeality. (They don't have lives.) It takes the coming apocalypse for them to realize that they have more in common with each other than with either the angels or the demons, and that they can rely on each other.
The performances were amazing. I particularly enjoyed Jon Hamm as Gabriel, Anna Maxwell Martin as Beelzebub, and Miranda Richardson as Madame Tracy (who was one of my favorite characters in the book), but everyone played their parts well. The Them were fantastic. I'm often hesitant about children in shows because the actors can be so hit-or-miss, but each one did well. The standout, however, was Michael Sheen as Aziraphale. Sure, David Tennant was also fantastic as Crowley, but honestly, it wasn't much of a stretch for him - the part was written for him, after all, and it was really a mix of the Doctor and Peter Vincent. Sheen, however, played the part of the fussy principality, a mix of indecisiveness and protective power, the angel afraid to get his hands dirty but strong enough to stand up to Gabriel (and honestly, the only person able to rein in Crowley). The show thrives on Tennant and Sheen playing off each other, but Aziraphale was the harder sell and Sheen nailed it.
Lastly, of course I loved the Doctor Who easter eggs. The first I saw was the SIDRAT license plate on the Young car. I think Aziraphale's robes had Gallifreyan embroidered on the shoulders but I could be wrong and I need to go check. I know I saw another but I can't remember what it was. (I am absolutely amazed there wasn't a police box on the street in the 1967 scene.) And when I saw the scene in my profile pic, I knew there would be a reference there, but I didn't want to stop and search each frame, so I went and got it off Reddit the next day. :) Oh, and people point out that when Crowley first enters the convent with the baby Antichrist in a basket and Mr. Young first seems him and mistakes who he is, Mr. Young calls him "Doctor".
So there you go, a shorter-than-I-really-wanted-it-to-be review of Good Omens. If you haven't seen it already, go watch it! If you don't have Amazon Prime, get the free one-month subscription, watch it, then cancel it.