Ok, now let's talk about Under the Red Hood as a complete story. The animated film differs from the comic in a number of ways, some better, some not quite as good, but because the better stuff is more noticeable, the film tends to be regarded as the superior product in the fandom. In fact, Under the Red Hood tends to appear not only on "Best Animated Batman Films" lists, but also "Best Batman Films" lists overall; it doesn't get quite the praise of Mask of the Phantasm or The Dark Knight, but is lauded the same. Easily the best change is having Jason Todd returned to life by Ras'Al Ghul, feeling guilty for leading Joker to Batman in revenge, dumping him in a Lazarus pit. In fact, the inclusion of elements from Death in the Family makes the narrative stronger overall. There are changes that irk me, or lessen the adaptation somewhat, but what they all tend to come back to is time; at a little over an hour, Under the Red Hood has to cut a great deal to fit everything in. Not that all cuts are bad, in fact, Batman figuring out the Red Hood's identity earlier makes for more dramatic tension, but there isn't much space for the story to "breathe".
For me, the annoying cuts are to the expanded characters, with generic power-armoured soldiers replacing Captain Nazi and Hyena and, most baffling, Black Mask recruiting the Joker to kill Red Hood. I get it, they already had a cast for Joker and he is the most famous Batman antagonist, but it just doesn't make too much sense for someone as smart as Black Mask to hire someone as unpredictable as Joker, in fact it backfires straight away and Black Mask almost ends up dead. Maybe Hyena and Captain Nazi were a bit too obscure for an animated film, but Joker still seems like a much worse choice than, say Tallyman, Deadshot or Clayface. Much like both versions of Superman II, the 'ideal' version of Under the Red Hood seems to be between both the comic and the film. I think Under the Red Hood could be a good candidate for a live-action film, so long as the script is kept tight and not too much time is taken up with Batman brooding over things or too much grunting. Also some female characters would be good, there's more in the animated film, but the imbalance is still pretty poor; maybe sneak in Spoiler, Catwoman or Oracle if making a big-budget film out of it.
Again, much like Superman II, both versions of Under the Red Hood kind of need to be experienced to get the better picture of what the story could be. Were I to be so silly as to attempt to put together a list of "essential" Batman media, both incarnations of Under the Red Hood would be on there, and it does form part of my "Head Canon". Red Hood, as a character, never really gets better than he is in this story either, the emotional intensity can't be maintained once the identity of Jason Todd is revealed and both he and Batman have confronted one another. The great thing about Red Hood in this story isn't how hyper-competent or edgy he is, but rather, that under all of the plans and skills, he's revealed to be an angry young man with a chip on his shoulder and one good point to make. The mythologising of the Batman family of characters is all too common in the fandom, and seeing a story make them all too human and flawed is what keeps me coming back to Under the Red Hood.