Deliberate mistake: In Natalie's bedroom, as Leonard is dressing, she sits at her vanity table, picks up a bright red hair pick and pulls her hair up with it. Just before kissing him, in the shot facing Leonard, the pick in her hair is lavender. (00:29:20 - 00:30:45)
Deliberate mistake: On the back of Natalie's photo the words, "She has also lost someone she will help you out of pity" differ. First, going into the restaurant, second, while in Natalie's bedroom and third, when he writes it at her house. (00:29:50 - 00:38:50)
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: When Leonard wakes up in Dodd's motel room, the positions of the pillow under him and the plastic cup on the nightstand differ in the close-up and following wide shot. Then after discovering Dodd in the closet, he goes back to his bed and the pillow, etc., change position again. (00:40:20)
Continuity mistake: When Dodd is in the closet, gagged with duct tape, the piece of tape reaches his cheekbones. Yet when Leonard rips the strip of duct tape to put across Dodd's mouth, it is considerably longer, long enough to reach his ears. (00:40:55 - 00:47:35)
Other mistake: In the scene where Leonard finds Dodd, he calls Teddy. When Teddy knocks at the door, Leonard goes to answer it. Just as he's about to open the door, there's a dodgy edit and you see Leonard jump to the left a little. (00:41:20)
Visible crew/equipment: Teddy, Leonard and Jimmy leave Jimmy's motel room and Leonard tells Teddy, "Take your own car." In the next shot as Teddy reaches for his door, a crew member, plus another one under his arm/elbow, are very clearly visible in the door mirror. Then as Teddy closes the door a crew member's face is seen on the left, in that mirror. (00:44:00)
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)
Visible crew/equipment: As Leonard stands in the bathroom looking at Dodd, unconscious on the floor after the two fought, visible in the reflection of the mirror behind him is a dark haired crew member with a blue shirt. It's in the same shot as the housekeeping woman that knocks. (00:47:15)
Continuity mistake: While Leonard fights with Dodd in the bathroom and after he locks the bolt on the front door, the towel hanging in the bathroom keeps changing position. (00:47:15)
Continuity mistake: When Leonard is looking at the note that reads, "Dodd white guy...get rid of him for Natalie..." in the wide shots as Leonard holds just the note, many photos are lying on the bed under his hands. Yet in the next close-up, the suit is spread out right under his hands as he holds the note with a single photo, that he soon drops, and no other photos are seen. (00:47:45)
Continuity mistake: After stuffing Dodd into the closet, Leonard holds the note that says, "Dodd white guy 6'2"...919 Booth St.," the note's handwriting is completely different in the wide shots and close-ups. (00:47:45)
Continuity mistake: As Leonard starts to write "Dodd" on Dodd's photo, in the wide shot that photo is lying with other photos on the bed and the cap of his pen is lying on the bed as well. Yet, in the close-up as he writes, "Dodd," on the front and "Get rid of him," on the back, the cap is on his pen, he writes the 'D' differently, there are no other photos and the suit is spread out under his hands. The next two wide shots differ between themselves too. (00:47:55)
Continuity mistake: The top of the driver's side window frame is clear of all the pieces of glass when he returns to the car after being chased by Dodd. Yet Leonard's car window, smashed from Dodd's bullet, has jagged large fragments still attached at the top of the frame, when he first leaves to be chased. (00:49:45 - 00:53:50)
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)
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: When Dodd's truck drives behind Lenny's Jag, in the long shot that features one car behind the other, it is clear that Lenny's (driver's side) window is down. When Dodd pulls up beside Lenny, the window is up, as it is when Dodd gets into Lenny's car. (00:52:45)
Visible crew/equipment: When Dodd pulls in behind Leonard in the alley the crew is reflected in his SUV. (00:53:00)
Factual error: On the upper right corner of the notes Leonard took about the police report, he wrote down the phone number of a detective (Brian Mezear?). It has 8 digits, intead of the 7 or 10 it should have. (00:57:10)
Continuity mistake: Since there are overlapping parts to the movie, certain scenes must be repeated. A segment starts with Leonard in bed at night waking up to a doorslam. When that scene is repeated, we see that he is woken up when the woman (escort) closes the bathroom door. The slam the second time (which really is the first) comes later than it did before. (00:57:25 - 01:02:30)
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.