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)
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)
Other mistake: In the scene where Leonard finds Dodd, he calls Teddy. When Teddy knocks at the door, Leonard goes to answer it. Just as he's about to open the door, there's a dodgy edit and you see Leonard jump to the left a little. (00:41:20)
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.