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: 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: 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 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: 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)
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 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)
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: 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: 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: 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: 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 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.