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: 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: 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)
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: 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)
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: 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)
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)
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)
Deliberate mistake: At the start of the film, when it's shown in reverse that Teddy lifts his head off the floor and his glasses rise up towards his face, in this particular shot it is very apparent that it is NOT Joe Pantoliano, who plays Teddy, that lifts his head off the tile. (00:02:25)
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)
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)
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)
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 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: 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)
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)
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.