Audio problem: In the first shot of the band inside the palace, to the left of the singer, there is a guy playing the drums. He's supposed to start playing the larger drum right before it cuts to the dancers. But, the sound of the drum starts before he starts hitting the sides of it.
Continuity mistake: At the start of the mine car chase look at the length of the braking poles on the side of the car. When they leave the main cave the roof lowers and the braking poles are slightly higher than the side of the car but were a lot longer to start with.
Continuity mistake: When that guy is lowered into the lava pit, the cage is level horizontally. When the empty cage is raised again, it's slightly at an angle.
Continuity mistake: During the mine cart chase, Indy hits one of the switches, putting it from the forward to the back position. When the guards in the mine cart get close, they shoot it, putting it from the front position to the back position again.
Continuity mistake: As the Thugee ceremony comes to an end, you see the stage hands exit and the spectators get up and start to leave. The next scene shows Short round looking over the edge and the stage hands are still on the stage, just starting to leave and some of the spectators are still in the process of standing up to leave.
Continuity mistake: When Willie walks out of her bedroom while Indy is fighting the assassin, the left sleeve on her pyjamas changes from rolled down to rolled up between shots.
Continuity mistake: In the mine cart, after Willie cries "Hurry" there's an aerial shot of the cart and Short Round has vanished.
Continuity mistake: When Indy is stopping the mine cart with his foot Willie's hand is on top of Indy's or not depending on the shot. This swaps back and forth.
Revealing mistake: When Willie runs into the death pit after pulling the release lever, when she drops the lantern Indy is not Harrison Ford but a stunt double. (00:59:30)
Continuity mistake: When Indy falls inside the mine cart, the protruding log disappears. A shot later it reappears, but when Indy hits the railroad switch with the shovel it disappears again.
Continuity mistake: Before the first sacrifice victim is being lowered into the lava pit, Mola Ram is standing on the edge of the pit, but in a few shots later he moves some feet away.
Continuity mistake: When Indy, Willie and the kid are leaving the village riding the elephants the sun changes between shots.
Other mistake: In the spike room, you see Shorty step on the switch that seals the two doorways. During that part, you see the stone extend from the wall, which he leans against to start the lowering ceiling. However, when Willie enters, without stepping on the switch, she pushes the stone to both seal the doors, start the ceiling lowering again, and extend the spikes. Why did it come out without anyone stepping on the switch beforehand like it did previously?
Other mistake: When Willie is given the pot of soup, she puts her spoon in and the eyes float up to the surface. Why are all of the eyes looking straight upward? Unless they're anchored in some way, not all of the pupils, if any, should be on top.
Revealing mistake: When Indy is put inside a cage Short Round comes to hug him. There's a shot of Indy from behind and he is a noticeable stand-in: Thinner face, clean hair, no sweat, and clean shaven instead of stubble.
Continuity mistake: After the man is burnt in the lava pit, Indy says the word 'diamonds'. Willy comes closer and then turns her head to the right. When the shot changes she's facing straight.
Continuity mistake: When Indy is fighting the Chief Guard on the conveyor belt, there is nothing behind the guard before he picks up the large rock, but in a close-up there are now small rocks on the belt.
Revealing mistake: When Indy is whipped six times, the first time we can see the whip doesn't touch him, or go anywhere near him.
Continuity mistake: When Indy hangs the assassin on the fan, in one shot he's spinning, but in the very next shot he's not. He can't just instantly stop, he'd have to slow down.
Revealing mistake: When Indy wraps his whip around an assassin's neck and hangs him on the fan, we can see it's not actually around his neck, it's just attached to his clothing.
Answer: This is conjecture, but it seems the general function of setting Temple of Doom before Raiders of the Lost Ark is that it helps set audience expectations that the two movies are self-contained episodes. For instance, Karen Allen has said she wasn't disappointed about not being asked to return because she'd already been told that the next installment was being set in the past before her character is reunited with Indy. Conversely, since we're already aware Raiders makes no mention of the events of Temple of Doom, we know we shouldn't necessarily expect any further installments to continue directly from prior movies' storylines regardless if they are set forward in time.
TonyPH