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)
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)
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.
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)
Deliberate mistake: When Leonard takes the photo of his newly acquired Jaguar, the car's hood is parked right up in front of the shrubbery and its shadow is in front, at the hood, seen when Teddy arrived. The actual photo of Leonard's 'new' Jaguar shows it to be nowhere near any shrubs and its large shadow is cast on its right side. (00:03:25 - 01:40:55)
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: In room 304 Leonard takes the papers that Natalie gave him out of the paper bag, he turns the bag over and written on the bag is "For Leonard from Natalie" with a line drawn underneath the words, It is written completely differently than when it was seen earlier as the waiter at the diner gives the bag to him when he came out of the restroom. (00:11:45 - 00:12:50)
Deliberate mistake: The reminder note that said, "City Grill Main St. today 1:00pm meet Natalie for info.," which Natalie wrote on a pharmacy bag, is seen in Leonard's hand at the Discount Inn just before he goes to meet her at the City Grill and in her bedroom when she writes and hands it to Leonard. The words on the bag are written differently, particularly the words "1:00pm meet Natalie for info." (00:26:15 - 00:30:35)
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.