Continuity mistake: In the scene in which Leonard writes on the back of his picture of Teddy, "He is the one. Kill him," we see a shot of Leonard placing the picture on top of his tan jacket on the bed as he goes to write on it. Then, it goes to a close-up of the picture in which the jacket has disappeared and the picture is lying directly on the blue bedspread. It actually happens twice within that same scene. (00:15:15)
Other mistake: At Natalie's house, when she is leaving and Lenny is taking a picture of her, the background looks totally different than from what is shown on the picture: the curtains are opened wider, the door is half open, there is a lamp on the left and there is no mirror on the wall. (00:17:10 - 01:20:50)
Visible crew/equipment: When Dodd pulls in behind Leonard in the alley the crew is reflected in his SUV. (00:53:00)
Continuity mistake: At the beginning of the movie, Leonard looks outside his window and sees room 304 (which will be given later to him on top of his current room) - it is on the same floor he is currently on. Still, when he learns from the clerk that he has been ripped off, we see them both coming out of room 21, which probably is room 221 with the first 2 missing, because it's next to the reception, itself located on the second floor (we know that because from room 304, Leonard only needs to go down one flight of stairs to reach it). Both his rooms, 304 and 221, could not possibly have been on the same floor. (00:06:40 - 00:26:25)
Trivia: In the scene where Lenny is on the phone describing how Sammy Jenkis ends up in a home due to the death of his wife, it shows Sammy sitting in a chair looking towards the camera. Sammy is temporarily blocked from our view by someone walking in front of him. As the person passes by, it is not Sammy that re-appears, but Lenny. To see it well, you will have to use the frame-by-frame mode. There are about 5 visible frames of Lenny in the home, and then it cuts back to him inside the motel room talking on the phone again. (01:29:55)
Factual error: In the beginning, Leonard looks at a copy of John G/Teddy's driver's license. The expiration date is listed as 2-29-01, but of course, 2001 was not a leap year, so there would've only been 28 days in the month. (00:12:45)
Deliberate mistake: When Leonard snaps a picture of Jimmy's body his head is lying between his shoulders looking straight up, but in the polaroid his head is lying to the right, and it is, in fact, a different actor altogether. (01:39:10)
Deliberate mistake: When Natalie writes the 'Dodd' note at her home, it just reads, "Dodd, MonteRest Inn on 5th St., room 6, put him onto Teddy..." However when Leonard reads the note in the car and at the motel, the note now reads, "Dodd, white guy, 6'2", blonde, MonteRest Inn on 5th St..." (00:50:00 - 01:12:15)
Continuity mistake: Leonard is twice shown looking at the back of the Ferdy's Bar coaster, with 'Come by after, Natalie' written on the back. The handwriting is noticeably different in the two shots, particularly the first two letters of 'after' and the middle 'a' in 'Natalie'. (01:20:15 - 01:28:55)
Deliberate mistake: Leonard shows up at Natalie's door and shows her the picture of Dodd. However when Leonard actually takes that photo of Dodd, not only does his face differ, the blood on his face differs from the photograph as well. (00:32:55 - 00:47:40)
Continuity mistake: The blood pattern splattered on the tile wall and grout, shown in the photograph that Teddy holds in the opening shot, differs from the pattern that is seen in the reverse shot while Teddy is still lying on the ground, before he even moves. (00:02:25)
Deliberate mistake: When Leonard first writes Teddy's number on his picture, he writes 555-1134. In the rest of the movie, it is written as 555-0134. The handwriting differs as well as the truck seen behind Teddy.
Continuity mistake: In the first close-up Teddy's eyeglasses are upside down, lying near the tile wall, the next wide shot it's upright, further away from the wall, next close-up it's near the tile wall again, lying on the grout just as it's moving in reverse, then in the next wide shot it's further away from the wall once again, lying on the tile, just before it reverses to go back up onto his face. (00:02:15)
Continuity mistake: When Teddy is shown getting shot in reverse, at the start of the film, Teddy's right arm position, Leonard's grip on Teddy's clothing, etc., are distinctly different than in the following scene when he is shown again being shot by Leonard. (00:02:30 - 00:06:15)
Revealing mistake: At the 'beginning' when Teddy and Leonard drive up to the abandoned building with the old truck out front, inside the truck Leonard handles a number of .357 revolver calibre bullets that have apparently not been fired, but you can clearly see the indented primers in the base of the bullets, indicating that they are just inert movie props. (00:04:40)
Continuity mistake: While driving, after escaping from Dodd, Leonard pulls out the 'Dodd' note from his pocket and holds it at the steering wheel with both hands. In the close-up of his hands he holds the photos, which face the windshield, in his right hand and the note in his left. However in the shot facing Leonard, he holds the photos, which now face HIM, in both hands and the note is GONE. (00:49:55)
Continuity mistake: When Leonard is running in the trailer park after having realised Dodd is the one who's chasing him, he makes a chair fall while escaping from Dodd shooting him. However, in the next shot, the chair returns to its place.
Deliberate mistake: The license plate on Teddy's car is constantly changing. When Leonard reads his card of the plate number, it could end either with 7IU or 71U since his I's and his 1's are written in the same way. When Natalie gives him the information, the car's registration ends in 7IU. Even the car's plate is both 7IU and 71U. Leonard's tattoo of the number ends in 71U, but he says 7IU. (00:44:17 - 01:46:55)
Answer: It is never explicitly given. The most Leonard says on the subject is: "I'd rather be mistaken for a dead guy than a killer." Speculations include (you can make up your own motives as well) : (1) The clothes and car are so much nicer than his. If you are willing to kill someone: stealing is not really a "crime." Why not take the nicer objects? (2) It could be part of his "routine": Kill a man, take his clothes and car. The clothes he had on and the truck may be from the man he killed a year ago. (3) It could be that he wants to make the killer of his wife suffer even more, and takes his clothes as a way of humiliating him. Leonard takes the man's life-his clothes and car, which are wrapped up in his identity-just as the man took his. This idea seems to work with a theme in Memento about "Identity" (especially mistaken identity). Natalie thinks Leonard is Jimmy, then thinks he is Teddy, then learns he is Leonard. Teddy is "mistaken" for the second killer, Jimmy is "mistaken" for the 2nd killer. Sammy's story as a part of Leonard's story, etc. (4) It could "simply" be explained as a "plot device": Leonard has to do it, otherwise he won't find the note in "his pocket" and meet Natalie. (5) Leonard doesn't want to admit he's a murderer. He's lying to himself. If he's the victim, then he cannot be the murderer. (6) Leonard takes Jimmy's clothing as part of his routine of killing J.G.'s he becomes another person, he's the victim not the killer, thus "I'd rather be mistaken for a dead guy than a killer." and that's why he also takes his car, so he has to, once again, find his wife's killer and kill him.