So: be warned. Spoilers follow. Don't read until you have seen the film (or if you never intend to see the film, or don't mind having key plot elements spoiled.)
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My daughter and I saw the film at a special pre-screening. There were probably about 60 people there. I thought they might be doing some A/B testing to tweak some of the scenes, but they didn't really seem to be monitoring the audience. In fact, the people running it appeared almost indifferent to the effort. They barely recognized that my daughter and I were dressed up as characters from the movie (granted, they were not the best costumes, but we were the only ones who did any cosplay) and didn't seem very familiar with the movie at all.
But the movie itself.
The opening scenes, from the wide pans over Zalem, the falling garbage, to Ido plucking out the recovered eye, up to the point where he holds the newly discovered Alita up in front of the setting sun are, almost, frame for frame, like the Manga. It reminded me a lot of parts of Sin City where you could tell they were comic panels. Fortunately, it stops there. I thought Sin City was way too slavish to the comic, to the point of being detrimental to the film. So after the initial shots, it sticks to a pure cinematographic perspective, which I think is a good thing.
It is when the differences start that things get interesting. Most elements than I expected are brought in from the Manga. They are often used in the same _role_ as in the manga, but are fit into the _plot_ differently. Like Hugo riding around on a mono-cycle. It's a cool visual from a different part of the manga, but works in this setting.
Completely new inventions are relatively rare. There are really only two major ones, and I'll talk about those. Mostly I felt they either enhanced the story, or else made it more consumable for cinema goers.
The first major invention is the role of Chiren. I completely loathed this character in the Anime. She seemed kind of pointless detracted from the "cast from heaven" motif contrasting Ido and Nova. I had no empathy for her and found her weak as an antagonist. She's still somewhat unlikable in the movie, but this is more due to character flaws than personality flaws.
(If you didn't take the warning seriously above, MAJOR SPOILERS do follow.)
In the plot of the movie, Ido and Chiren were husband and wife (vaguely hinted at in the Anime) on Zalem. They had a daughter, but the daughter was disabled. (Legs paralyzed.) Instead of having just the daughter cast out from Zalem, they accompanied her and made a life in the Scrapyard. She was really good at biology bits, he was really good at cybernetic bits. They mostly made their money building super competitive motorball players. With the money they earn Ido makes a beautiful and fast body to replace his daughter's paralyzed one. But one of their "motorball creations" goes crazy, breaks into the clinic looking for drugs, and in his escape kills their daughter. This creates an irreconcilable split between Ido and Chiren as they respond to the tragedy differently. Chiren lives in denial, will do anything and everything she can to escape it, and return back to Zalem. Ido accepts it, but has difficulty processing it and letting go.
So although a significant departure from the Manga, it sets the stage for a lot of things to be used from the Manga but with a tighter plot.
The tattoo filigree arms Ido first gets for Alita in the manga becomes the body he made for his dead daughter, which he was supposed to throw away but did not because he didn't let go.
Ido gives Alita her name not from his cat, but from his daughter. Which makes it all much more poignant.The main tension in the Manga between Ido and Alita was her self determination against him wanting a dress up doll. This pivots that to being more a coming of age father/daughter thing. Much more tender, understanding, and showing good roles as he moves from sheltering her, to supporting her, to mentoring her. (Of course, I saw it with my 11 year old daughter, which means this probably hit home a lot more deeply with me than it might with others!)
One thing I thought they would drop for sure was the whole "Ido is a hunter warrior because deep down he gets a certain thrill killing people". But they kept it, in this form, in the movie. Ido became a hunter warrior to track down the motorball freak he had made that killed his daughter. He felt responsible, and as time passed and he couldn't find him, he felt responsible for all of them. He turned to vigilantism to assuage his grief at what he had done and swore off never to build freaks like that again. (Setting up a contrast with Chiren.) So I was impressed they kept this aspect of his character and even enhanced it.
So, overall, I'm OK with this departure. It allowed for many other elements to be used, and I think even enhanced the plot.
The next major departure as to do with the back story. In the movie, Mars, for no explained reason, attacked all the "floating cities" of Earth, destroying all but Zalem. This precipitated The Fall and a backsliding of technology. Alita was part of a martian brigade of bezerker body warriors sent to take down Zalem. She has flashbacks training for zero G fighting, fighting on the moon, and then assaulting Zalem via a suspension cable. In this last flashback one of those spinny-blades-of-death thingies comes down and disrupts their unit, knocking her off. It's sort of implied that's how she ended up in the scrap heap, but that leaves open a big hole since it doesn't explain how her head/core got separated from her body. (In the movie she finds the bezerker body in a nearby crashed spaceship, nicely parked in its charging station.)
So, yeah, I kind of get it. Kishiro can put "Oh, yeah, Triphares, it's a floating city, we don't go there" and leave it sit for a major chunk of the Manga, but you can't do that in a film. People want backstory so they can connect with the plot. And it might be hard to have a backstory closer to the Manga because it is either insanely complicated or we still are finding out what it was all about. On the balance of things, it doesn't affect the plot of the movie that much, so I'm ready to give it a pass. I do think it will affect the plot of any sequels, which is disappointing. But I'll withhold judgement until we see if sequels are going to be a reality or not.
On to other departures.
Borderline "major" is the role of Desty Nova. He's still a mastermind pulling the strings on all sorts of things. No specific mention of Karma. More that he's "immortal" and finds it amusing. It's hinted at that the Mars assault was specifically to kill him. Maybe some overlap there between Nova and Trinidad. But it's going to affect any possible sequel more than this movie, so I'm not so bothered with it.
There is a character named Koyomi in the Manga. There is a character called Koyomi in the film. Other than being female, they have nothing in common. I like both though.
The Damascus Blade isn't very damask. It's also purportedly "martial tech" and Alita seems to be able to route her plasma through it. She doesn't use it much, since Zapan has it most of the time. And there is a lovely shot at the end where she holds it in reverse grip fashion like we see in the Manga. It's just a pose, so it's solely there for the Manga fans!
Zapan does not die. Which sets him up as an antagonist in any sequel. He's got a lot of good dialog in the film.
Vector does die. Which is a bit of a shame, since I think it was unnecessary.
Hugo is, well, Hugo. And is probably the least changed character from the Manga. I didn't like him that much in the Manga, thought he was a dork in the Anime, but he's OK in the film.
Beyond plot...
The cinematography is REALLY well done. After the screen was some "behind the scenes" stuff, and a "live" chat with the producers, directors, and main cast members. They emphasized a lot that it was all "performance capture" rather than "motion capture", which normally I would just take as blah-blah-blah. But, really, it was. Alita never looks or acts "plastic". Like in that horrid Final Fantasy film they zoomed in at points and I'm like "oh, look, they rendered her pores too". But in this movie I was too capture to be looking for CGI flaws. The character is really ALIVE. Reflecting on it there were scenes where she's wearing bulky sweaters to cover her cyborg body. Normally you see that to save on CGI so you can use the normal actor. (Like having characters in Anime speak through megaphones so they don't have to animate the mouth.) But it's ALL CGI, so they were doing that for the character.
One point they made in the talk at the end that while the Alita character was CGI, the bulk of the sets were real. They, apparently, only did 3 days of green screen filming. Obviously somethings like when they run over skylines or leap between buildings, and backdrops, are CGI. But anything close is real. I agree with them that it helps the actors give a more genuine performance (contrast with Phantom Menace) and just makes it "feel" more real.
The performances were really good. Especially Alita and Ido. There's a lot of sensitive father/daughter stuff going on there and they capture it very well, making it real for the audiences. I must have cried about 8 times, but, then, I cry at old episodes of Star Trek.
So take it as you may.The other characters were mainly there in supporting roles. They did good at that, which is all I would ask of them.
Overall I really liked it a lot. I expected to like it. But it ended up being better than I expected. But I also liked the Lord of the Rings movies. I expected an adaption, faithful to the intent of the original, but not a literal reconstruction of the original. I was willing to cut it more slack than I did, so was pleasantly surprised with how faithful it was.
After reading your review, I would cry watching that movie too, but for very different reasons. 
