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: 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)
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)
Continuity mistake: There are scenes in which Leonard is looking at the back of Teddy's polaroid. If you look at the serial number of the polaroid when he is writing "Don't believe his lies" it is a different number than when he looks at it during the beginning of the film. (00:05:15 - 00:07:00)
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)
Continuity mistake: The words behind Teddy's photograph, "Don't believe his lies" are written differently on the same photo three times. First, when Leonard reads it at the abandoned building, second, when he writes it and third when he looks at the photo after seeing a copy of Teddy's license while in room 304. (00:07:00 - 00:13:05)
Continuity mistake: After entering room 304, Leonard steps on the bed to put the 4 photos back up on his paper on the wall. In the 1st close-up he puts the Discount Inn and the 2nd close-up he puts up Natalie. Then in a wide shot of Leonard looking at the paper, visible at the top of the paper, in the center, is the photo of Natalie and on its right is the photo of Teddy, already up. In the next shot, the 3rd close-up, he puts up his car. However in the next wide shot ALL three photos that he just put up in the close-ups are GONE. Then in the 4th close-up he puts up Teddy's photo, which was up in the first wide shot. (00:12:25)
Continuity mistake: Under the photo of Natalie, the name Natalie is written. When the photo is seen in room 304, on the wall, then when he finds it while in her bedroom, and when he writes it at her house, the handwriting differs. (00:12:25 - 00:38:35)
Continuity mistake: Leonard sits down at the table in room 304 and starts to pull out from the bag the info that Natalie provided. His keys are positioned differently on the table in the wide shot and close-up. (00:12:45)
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)
Continuity mistake: When Leonard and Teddy are in the diner, after Teddy asks, "You think he's still here? Jon G., the guy you're looking for?" the amount of water in Leonard's glass changes in the wide shots and close-up. (00:23:25)
Continuity mistake: At the diner after Teddy asks, "Where are you staying?" he wipes his mouth with the napkin. The position of the napkin in Teddy's hand differs in the shot facing Leonard and the shot facing Teddy. (00:24:40)
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)
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)
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.