Continuity mistake: The raft arrives in a big place with no snow, yet from the opposite front angle there's snow all around.
Continuity mistake: When Willie falls from the elephant into the puddle of water, her hair varies from very wet at the bottom to mostly dry from each different camera angle.
Revealing mistake: When Indy is hanging on the cliff as it falls out from underneath him (right after the mine cart scene), as he's kicking, you can see the stone wall moving in and out as he hits it after the rocks fall out underneath him.
Continuity mistake: In the mine cart, when Indy hits a tank with a shovel, the Thuggees behind raise their arms, yet a frame later, from a wider angle the arms are lowered.
Continuity mistake: When Willie and Short get to the bridge there's a big tree on the left. In subsequent shots it's gone.
Continuity mistake: After the pilots bail out, the main engine has already stopped with the prop blade between five and ten degrees off-center, angled right. In the shot before Indy and company bail out, the prop is angled left, then right again.
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.
Continuity mistake: On the ladder, when Mola Ram tries to grab Indy's heart, Mola's hands and Indy's chest are very sweaty and dirty in the close-ups, but dry and quite clean in the wide angles.
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