It is claimed by Cyborg that the mother box came to life after Superman was killed in the events of Batman vs Superman, however in the video that Wonder Woman watches during BvS it can be seen as reacting and creating Cyborg.
In the Tunnel fight scene, before Flash starts saving civilians, Flash states how he is afraid. This should have attracted the parademons since they can smell fear, but given multiple civilians much closer to parademons and much more afraid than him, it's perfectly reasonable to assume that their fear was overwhelmingly stronger and completely masked Flash's fear.
At the end of Batman v Superman: Dawn of Justice we see dirt rising up from Clark's casket, indicating his return from the dead. In this movie that scene seems to be ignored, since Clark is clearly dead when they dig him up. Evidently, the dirt is not rising because Clark is coming back to life on his own. It could be some residual effect of his powers (kind of like a human corpse twitching) or something else entirely. In any case, this scene was clearly intended to be a teaser for the sequel and nothing more.
When Batman is driving the bat-tank and One of the flying aliens rips the cannon off of the bat-tank and starts to smash the bat-tank with it, however later when Aquaman is standing on the bat-tank the cannon is still attached. Actually the creature tore off one the smaller machine gun barrels to destroy the other one. The bigger cannon that Aquaman is standing next to fires much heavier ammunition and was never destroyed.
Near the beginning, when Wonder Woman uses the "Lasso of Hestia" to compel a man to tell the truth about who he and his compatriots are, he replies "we're a small group of reactionary terrorists". Terrorists usually think of themselves as freedom fighters, patriots, warriors of God etc., not terrorists. However, not every terrorist is a true believer, becoming a member of a terrorist group for other reasons (greed, revenge, confusion, coercion, etc.), and even true believers can feel the error of their ways deep down, not even admitting that to themselves until the Lasso compels them to.
When the Flash character goes undercover to sneak into S.T.A.R. labs, he is wearing an Air Force uniform with one stripe on his sleeve. He then hands his military I.D. card to the guard who then scans it. After scanning, the computer screen displays information indicating his pay grade is E7. An E7 in the Air Force would have six stripes on his sleeve, not one stripe. However Victor who is hacking the system to grant Barry access is still figuring out his powers and as the team are on a time limit it's so he just picked a random rank that would allow Barry to enter a top secret base.
The Parademons are shown to have super strength (as, for example, they ripped off parts of the Batmobile and they easily caused it severe damage), but they were attacking the Russian family's house (which has a very poor structure) for a long period of time and they couldn't even storm in.
When Barry and Cyborg go to Kansas they are both wearing hoodies and Barry is not wearing his speed suit. Unless Cyborg carried him, Barry could not have gotten to Kansas without a costume and with his street clothes intact.
After Superman's resurrection, the Mother Box is thrown out from the Kryptonian ship and it lands on a car. A few moments later, a Boom Tube appears on the Kryptonian ship and Wonder Woman says "Steppewolf has the Mother Box". But the Mother Box was never on the Kryptonian ship.