Continuity mistake: At Club Obi Wan, Indy places Nurhachi on the table in the closeup with his left hand, but when Indy puts the diamond on the table with his left hand it's not actually Indy's hand that we see in this shot.
Continuity mistake: During the melee at Club Obi Wan, Lao Che's goons toss Indy onto the dish cart which has a horizontal metal bar between two of its legs over the wheels, but in the closeup as Indy tries to grab the antidote vial, the metal bar has vanished, but it reappears when it crashes into the bandstand.
Continuity mistake: When the pilots jump from the airplane we see Willie asleep directly under the rolled up tapestries, but when she wakes up and starts to make her way to the cockpit as she coughs, in the wideshot we see she's coming from the area further back toward the tail.
Continuity mistake: When Willie and Shorty get to the rope bridge they start walking out and Shorty starts jumping on it, then in the next shot the sunburn on Willie's back is significantly worse and her hair differs. In the second shot it's presumably her stand-in.
Continuity mistake: While hanging onto the fallen bridge, both Mola Ram and Indy are pulling Indy's bag with the Shankara Stones, and note that this bag has three drain holes underneath instead of just two drain holes, which it normally has.
Factual error: The image of the plane's instrument panel shows a VOR, just to the left of the orange circle with the triangle in it. The VOR is "a type of short-range radio navigation system for aircraft, enabling aircraft with a receiving unit to determine its position and stay on course by receiving radio signals transmitted by a network of fixed ground radio beacons. ... Developed in the United States beginning in 1937 and deployed by 1946..." (Wikipedia). The movie's opening scene is Shanghai, 1935. (00:16:45)
Continuity mistake: During the palace dinner scene, the "snake surprise" is laid down at Willie's left side. A few shots later it's on her right.
Continuity mistake: While Indy and the Thuggee Chief Guard are battling it out, after the shot of the Maharaja gleefully stabbing the Indy doll, in Willie's next closeup she's wearing the veil as she throws rocks at the Chief Guard, but in the next wideshot the veil is gone, but it's back on her head in following shots.
Continuity mistake: Mola Ram wears a wristband on each wrist and only one armband on his right upper arm, but when he falls off the hanging bridge into the water, in the overhead shot he's suddenly wearing armbands on both upper arms.
Continuity mistake: After Willie's rescue, when she tells Indiana they should get out of there, in the next closeup just as Indy says, "Right, all of us," this is a flipped shot, note Indy's Fedora, chin scar and red strap.
Continuity mistake: During the mine cart chase, when Indy grabs the shovel to scrape the underside of the overhead metal bin, Indy's right hand is suddenly missing the cloth which is wrapped around his knuckles in previous and following shots.
Continuity mistake: While the second mine cart with Thuggee goons is in hot pursuit of Indy, note the brake pole at the front of Indy's mine cart keeps vanishing and reappearing in following shots, before it actually breaks off.
Continuity mistake: After Indy's fight with the Chief Guard on the conveyor belt, when Indy attempts to save him from the rock crusher Indy ends up hoisting himself up in mid-air, and when he reaches the platform Indy's shoes are completely flat, unlike Indy's ankle boots which have heels.
Continuity mistake: When Lao Che's goons toss Indy onto the cart at Club Obi Wan, note the soles of Indy's shoes are entirely tan with black heels, but when the cart crashes into the bandstand we see the arches of Indy's soles are black and the heels are tan.
Continuity mistake: After Indy and Willie have jumped out the window at Club Obi Wan, when they land on the lower awnings and roll off we can see Willie's high-heels have become flats.
Continuity mistake: After Indy, Willie, and Short Round jump out of the plane with the raft, when the raft finally slows down in the water Willie falls backwards with her legs up in the air and her high-heels have become flats.
Factual error: They escape from the plane using a life raft and fall a considerable distance (looks like hundreds of feet) before landing on a snowfield. They then slide down the snow and off a cliff before landing in a fast flowing river, again falling a considerable height. There is no way they would be able to fall that distance and land on snow covered ground or a river without being killed or seriously injured, but they are completely unscathed.
Factual error: The self inflating boat on the "South China Air" plane is the typical modern colour yellow and modern shape, which didn't exist at that time.
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.
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