First, a caveat: Adaptations from one medium to another are, as they say, fraught with consequences. One of the most successful adaptations I've seen was Brokeback Mountain, and that was developed from a short story. One might guess, then, that adapting a trilogy of novels that run well over a thousand pages would pose some risks. I've tried to stay away from comparisons, because the film is not the book, but there are a number of problem areas that caught my attention that are, I think, internal to the film.
Character and motivations: What I call the "basis" for the action. A few things jumped out at me: Merry and Pippin wind up on the quest by happenstance, and there's really no foundation for their participation aside from that. We get no real sense of a bond between them and Frodo that would support their involvement on any other than the most rudimentary level. Likewise, the relationship between Eowyn and Faramir is left to assumption -- we get one glimpse of them standing next to each other at the coronation, but as I recall, there's not even a glance to seal the assumption. Aragorn as a character is well done, within a somewhat limited scope, but there's a dimension missing: what he is in the movie is an adventurer on his way to becoming a king and not a king on the way to reclaiming his throne, despite repeated references to his heritage. There's a difference. Leaving the reforging of Anduril until late in the story was a mistake, I think, as was glossing over Aragorn's confrontation with Sauron via the palantir. Again, it leaves important parts of the story with no basis -- too often, I found myself wonder "Why is this happening?"
I don't know what Cate Blanchett had been smoking -- she looked and acted like she was stoned -- but it should probably be illegal. Galadriel has none of the humanity one sees in the other elves, and again, it undercuts the impact of what she says and does.
On the whole, the characters for the most part lack depth, and given the caliber of the cast, I have to fault the script.
The one other important respect in which I think the film trilogy falls short is that Jackson couldn't seem to settle on a consistent tone. Even without reference to the books, what we're seeing is high heroic fantasy, in which we're assured the good guys win (Jackson even sets it up so that the Shire is completely untouched by Sauron's machinations), but the context is Hollywood realism in a fairly run-of-the-mill adventure story. It's an uneasy mix, and Jackson didn't seem to be able to find a comfortable middle ground.
The Fellowship of the Ring is certainly the best of the three -- the tightest and most coherent -- while the other two gradually come unglued.
I'll probably watch the whole thing again, at some point, in spite of Jackson's tendency to milk scenes that need to be tight and truncate scenes that need more development -- there are too many of those to enumerate, but take the final scene in the Grey Havens as the type specimen -- because there is credible work done by the cast, and the scenery is gorgeous. (And there's another disjunction with the action: there's more fantasy in the landscape than in the story.)