(at around 55 mins) When Bella finds Jacob in the rain, she slams her car door and it doesn't shut fully. The next scene you see the car shut completely.
When Bella is in the kitchen telling Charlie and Harry Clearwater about seeing the huge wolves in the woods, there is clearly no phone on the wall between the cupboards and the doorway - however towards the end of the film, Jacob answers Edward's call using a phone on the wall, that wasn't there in the earlier scene.
(at around 1h 5 mins) Right before Jacob jumps through Bella's window you can clearly see that her desk is pushed up against the same window but when he jumps through it, the desk is moved to a completely different spot in the room. Once he jumps out again the desk is moved back to its original location in the next scene.
Esme's hair is clearly shown in the first Twilight (2008), as was also the case in the books, to be caramel colored. In New Moon, Esme's hair continuously switches between caramel and black.
(at around 1h 6 mins) When James bit Bella in Twilight (2008), his upper set of teeth marks are in the center of her wrist, with the lower set being on the side of her wrist. When you see Bella's scars in her room with Jacob, the entire set of teeth marks have been shifted markedly to the right.
(at around 44 mins) When Jacob shows Bella the brake on the motorcycle, he shows her the front brake (right hand), not the rear brake (right foot). When at speed, if you use only the front brake it will flip the bike. If he had actually known what he was doing, he would not have done this.
(at around 1h 22 mins) When Jacob is giving Bella CPR on the beach, when she is revived, she doesn't cough, nor does she spit up any water. If she had, in fact, drowned, her lungs would have been full of water; upon revival, she would have coughed up the water in her lungs.
(at around 54 mins) When Bella is trying to reach Jacob after his abrupt departure from the movie theater, she is in her living room doing her homework while her father is watching an Arizona Wildcats college football game on TV. Football season would have been over by this time, which should have been March or April (according to the fact that Bella's last visible email to Alice was on February 13, her sent email folder is organized by date and the last emails she's sent are at the bottom, as evidenced by the fact we see February 11, yesterday and today at the bottom) and she had spent at least a month or so working on the bikes with Jacob).
(at around 1h) Laurent doesn't glow like the vampires do, though he is outside. However, it's not so much sunny as overcast. We see Edward and the other Cullens outside many times during the day when it's light but overcast, and they never sparkle.
(at around 16 mins) Since Carlisle has been a doctor for over 100 years, he could still have plastic syringes with metal cap needles for personal use at home.
At Bella's birthday party, all of the Cullens except for Esme are pale. Esme has the skin tone of a human being.
(at around 56 mins) When Bella is talking to Jake in the rain, only the front of her jacket is wet.
(at around 42 mins) When Bella and Jacob are driving to ride the bikes, as they are being filmed in the truck's cabin, the shadows from the road trees are passing like they were driving backwards instead of driving forward.
(at around 4 mins) After Bella's birthday, which is when she receives a digital camera, she takes her camera to school and takes a picture of her friends. The image in the view-finder does not match with the scene being photographed. In the scene, there are two people walking behind her group of friends that don't appear in the view finder.
(at around 1h 45 mins) During Edward and Felix's fight, the harness Edward is wearing under the robe is visible in some shots.
(at around 1h 50 mins) In Bella's bedroom, after Edward comes back, when Edward asks if Bella could forgive him, Bella is heard saying "come here" but her lips never move.
(at around 7 mins) In the beginning of the movie, when Jacob is saying "Happy Birthday" to Bella, his voice doesn't match with the words.
(at around 1h 30 mins) When Alice and Bella are flying in the plane to Italy to save Edward, on the back of the plane it says "Virgin America", which does not fly to Europe. They would need Virgin Atlantic to fly to Europe.
(at around 1h 45 mins) When Edward is fighting with Felix and is grasped by the neck by Felix and subdued, his chest and torso is shown to be heaving, out of breath from the exertion. However, vampires do not need to breathe.
During the birthday scene when Bella cuts her finger Jasper freaks out over a little drop of blood. But in the first movie when all of the Cullen's are rescuing Bella from James, Bella's femoral artery is severed and she's gushing out blood. You can clearly see Jasper in the background holding James while Edward and Carlisle are helping her. With all of that blood Jasper has no problem with it.