Revealing mistake: The skull which Indiana forces under one of the wheels in the spiked room's ceiling has a very straight, clean-cut line through the cranium. It has none of the imperfections or signs of growth which would occur naturally; this is a very fake-looking skull.
Audio problem: A few times during the banquet scene, various snakes make noises like squealing pigs or large bugs.
Other mistake: During the Tri-Motor crash sequence, the crew dumps the fuel from the aircraft and bails out. In the shot of the red fuel pouring out, the fuel is shown streaming out of the exhaust pipe for the center engine, which would be hot enough to cause the fuel to ignite and either set the aircraft on fire or cause it to explode.
Continuity mistake: During the mine cart scene, Indy hits the second lever with the shovel to change the course. The Thuggees in the cart behind him shoot the same lever and it flies back just as Indy hits it, and changes the course again.
Other mistake: In the "tunnel of bugs" you may reasonably expect there to be such creatures as cockroaches, spiders or giant centipedes, but the majority of the insects are locusts which are herbivorous and would not be found in an underground tunnel. Other out-of-placers are giant stick insects (also herbivorous) as well as a South American harlequin beetle (this part of the film is set in India). You can also see, in the scene where Willie turns her hand over to reveal one of the giant stick insects, that she is cupping it to hold the insect in place.
Audio problem: When Indy, Willy and Short Round are leaning on the mountain where the jet of water comes out, none of their cries match their lips.
Revealing mistake: In the wide angle of the mountain where Indy, Willy and Short Round lean on, they cast no shadow, revealing it's a composite image.
Continuity mistake: In the angle from below where Short falls through the hole on the bridge, there are no ropes hanging off the sides. In all subsequent shots there are many ropes.
Continuity mistake: The dozens of balloons next to the gong base suddenly disappear when it falls down.
Revealing mistake: When leaning against the mountain, in the wide shot Short's face turns totally black, Willie becomes semi-transparent, and Indy's chest becomes greyish colour, all because of the blue-screen effects.
Revealing mistake: When the yellow raft slides down the snow, in the close-ups it turns white because of the blue-screen effect.
Revealing mistake: When the rickshaw crashes, neither the driver nor the girl move at all. Instead the sailor stands up by his own means and jumps, which is a really bad stunt.
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: 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.
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