Continuity mistake: In the first sequence in which Leonard arrives at the Discount Inn for the first time he gets out of his car, which is parked on the street, then goes down a ramp to enter the inn. In the next shot he is suddenly across the street, photographing the inn's sign. Then in the next shot he is getting out of his car, which is now parked somewhere else at the inn, looking at the photo he just took. (01:00:10)
Continuity mistake: In the first scene where Leonard is asleep and the prostitute goes to the bathroom and slams the door, Leonard is facing away from the camera, the door slams then he puts his arm on the other pillow and rolls over. When this scene is repeated a short time later Leonard is facing towards the camera when the door slams. (01:00:10 - 01:02:35)
Continuity mistake: While Leonard sits on Natalie's couch rifling through his papers, in the shot facing Leonard, behind him, the two top drawers of the writing desk are open. In the next shot as Natalie comes home only the top drawer is open. (01:13:20)
Continuity mistake: While Leonard sits on Natalie's couch looking through his papers, in the shot facing Leonard, behind him, the top two drawers of the writing desk are open. In the next shot as Natalie comes home only the top drawer is open. (01:13:20)
Continuity mistake: When Natalie comes home she slams her handbag down on the shelf. As she and Lenny talk and she begins to antagonize him, the handbag changes position within these shots. (01:13:25)
Continuity mistake: The top of the writing desk has different items on it, including a ceramic pot with pens. When Natalie comes home she takes the pens out of the pot and slips them in her handbag. The existence and/or position of the things on the top of the desk change within the previous and following shots of it, including when Leonard searches for a pen. (01:13:25)
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)
Other mistake: In the Sammy/Lenny shot in the Institution, before the switch the orderly on the left heads toward the medical files with his arm outstretched and the nurse walks behind Sammy. Just as the person passes the camera the film jerks and the two men at the table, visible on the left, have a shaking motion. After the switch, the orderly and the nurse are both further back. (01:29:50)
Continuity mistake: When Leonard is getting the tattoo of the license plate on his leg in the tattoo parlor, we see a closeup of the tattoo artist writing the "n" in "license". Then it cuts to a farther shot of her writing. It buzzes for a little bit, and the sound corresponds to her just finishing the "e" on "license" when Teddy walks in and she stops. After the dialog, when the tattoo artist starts again, she has magically finished the tattoo, finishing the sentence "license plate number: SG13 71U" in the period of about 2 seconds. (01:30:15 - 01:31:00)
Continuity mistake: Leonard removes his large 'photo' paper from the wall and places it on the bed to fold. Things are positioned differently in the close-up and wide shots. (01:34:20)
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)
Continuity mistake: When Jimmy is killed, his body lies on a certain area of floor tile with missing tile, yet after Leonard takes the photo, there is a close-up of Jimmy's body and it lies on a different area of the tile, further up, closer to the wall. (01:39:15)
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: 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.
Continuity mistake: The window in Dodd's motel room is on the right side from the outside. When Leonard enters, it's on the left side. Christopher Nolan acknowledges this mistake on the DVD commentary and even says that no one's noticed it yet.
Continuity mistake: After Lenny woke up on Natalie's bed and was leaving and Natalie kissed him, her lip's deep red injury disappears and reappears randomly.
Continuity mistake: Leonard is several times 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 various shots/scenes, particularly the first two letters of 'after' which are capitals in some shots, and the middle 'a' in 'Natalie'.
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.