On the DVD commentary of "The Lord of the Rings: The Fellowship of the Ring," Peter Jackson acknowledges one shot, a low angle of a hobbit at Bilbo's birthday party shouting "Proudfeet!", as an intentional homage to Bakshi's film. By far the biggest "lift" however is the scene of the Nazgûl appearing in the room at Bree and slashing the beds to ribbons thinking the shapes under the sheets to be the hobbits. This is almost identical to Bakshi's version which is significant as the scene is not depicted as described in the book: in the book, the attack is carried out by the Nazgûl. Some of Sam's interjections are also sourced from Bakshi rather than Tolkien.
In Rivendell when you first see Bilbo he is stood reciting poetry, while the other four hobbits walk into the room. In one frame you see Frodo sat in a corner playing a harp, then it returns to him entering with the other hobbits and it is an elf playing the harp.
In the first shot of the orc attack in Moria, Legolas' bow is tall with spiral tips (which he uses throughout the movie). In the very next shot, two seconds later, it is much smaller, with the regular curved tips of a straight-limb longbow.
United Artists insisted on the name change from Saruman to Aruman to avoid confusion with the similarly-named villain Sauron. The decision was later reversed during production, thus the brief references to "Aruman".
Q
Why wasn't it indicated that this would be the first part of the story?
A
The original title was "The Lord of the Rings Part One." After the film was completed, the studio told the director that no one would pay to see half of a film. The director objected to the name change, because it would confuse audiences into thinking that they would be getting the whole story. Nevertheless, the studio went ahead and issued the film as J. R. R. Tolkien's The Lord of the Rings, against the director's objections.
Q
Why does the film only cover half of the story?
A
The director had originally proposed a three-part adaptation to United Artists, the then-current rights holders. UA negotiated down to two films, and later refused to fund a sequel.