Continuity mistake: A large pile of cash mysteriously appears on the table in the nightclub scene when Indy asks for the diamond. The glass nearby also refills itself. (00:04:15)
Continuity mistake: Near the end when Kate Capshaw, Harrison Ford, and the kid are on the bridge, Harrison cuts the bridge with a large sword. Several men fall into the water below. The bad guy grabs hold and hangs on to some of the henchmen. The number of henchmen changes between shots. Close-up 3, wide shot 2, close-up no men, then back to 2 and so on. The bad guy throws some off as he climbs up the bridge but the number changes in the wrong order.
Continuity mistake: When Indy fights the two guards before the bridge scene, the sequence is all off. He knocks one to the ground, who lands still clutching his sword. The other one has his sword whipped away from him - it flies over the cliff, and he runs off, whimpering. We then see the other guard approach, and suddenly Indy's standing over an unconscious guard by the boulder and clutching a sword, both of which have appeared from nowhere. (01:37:25)
Continuity mistake: On the bridge scene Indy's sword changes. Initially, he has a sword with a straight blade and no hilt; however, when he shouts at Mola Rom, he has a something like a cutlass curved blade and heavy handguard to a machete-like weapon he uses to cut the bridge. (01:38:05 - 01:38:55)
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.
Revealing mistake: After Indy escapes from the conveyor belt, he runs on some suspended planks while being shot at. Watch the planks: There's black masking tape where the explosives are (which disappear in later shots). Furthermore, right before he takes out his whip, he grabs a railing where the bullet has just hit, and you can see that the black tape is dangling loose after the spark.
Revealing mistake: During the mine chase, after Indy cries to Shorty "Let it go, our only chance!" the wig of the doll replacing Willie flies away. (01:35:59)
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.
Revealing mistake: When Indy is hanging off the bridge and is punched by Mola Ram, Indy's safety wire attached to his belt is very noticeable. (01:47:45)
Continuity mistake: Before Mola Ram hides through a trap door on the floor, his hat is off except for a very brief moment when it's on.
Continuity mistake: When the pilots jump off the plane the flowers on Willie's hair swap positions for a brief moment.
Continuity mistake: After Short Round breaks the Maharaja's spell, the blood on the side of his mouth swaps from a thick strand to a thin one between shots.
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.
Visible crew/equipment: When Willie is brought the soup, many stage lights are reflected on the tureen, and the guests seated in front disappear.
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.
Continuity mistake: Mola Ram falls off the bridge-ladder when a step breaks in two, but from the immediate wide angle it just moves slightly out of place. Furthermore, in the next angles the step is perfectly fixed.
Continuity mistake: When the three heroes arrive at the village there's not a single bit of grass around. When they leave the place the morning after there's vegetation everywhere.
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