The Apollo 11 launch site is referred to as Cape Canaveral. While this would have been natural for Agent J, it would have been unnatural for young Agent K as the Cape was re-named Cape Kennedy after President John F. Kennedy's assassination in 1963. The name only reverted back to Cape Canaveral in 1973, way after Apollo 11 was launched.
The Spanish flag flapping in the wind during the Cape Canaveral launch scene is the current Spanish flag. In 1969, Spain - under the Francisco Franco regime, had a different flag (bearing a coat of arms with a black eagle).
The 1969 cops wear light powder blue shirts introduced in 1973. The correct uniform would have been dark navy blue shirts without NYPD patches on the sleeves, which were not designed as the department emblem until 1973 as well.
In 1969 every baseball cap had a green underbill. In the film you can see a 1969 Mets pitcher using a gray underbill cap. Gray underbill caps aren't introduced until 1994. In 2007 this color was dropped in favor of the current black.
During the Andy Warhol party scene the stage curtains are supported by an aluminum truss structure. In 1969 this type of system did not exist and any truss from that era would have been made from steel.
The slidewire astronaut escape baskets were not installed until after the Apollo era, and were used for the Space Shuttle. Apollo astronauts were expected to descend in a fast elevator, then retreat to a bunker under the pad in case of emergency.
Also during the Andy Warhol party there are lights seen on the stage curtains, commonly called a Star Cloth or Curtain today. This technology as displayed in the film did not exist in 1969, in fact any sort of mass production of very low powered Red LEDs only started in 1968 and prior to that most LEDs cost in the vicinity of $200 each!. White LEDs did not exist then.
The Colonel wears a brown belt with a brown holster. In 1956 the US Army officially changed the color for belts, holsters, and shoes from Cafe Brown to Black. In 1969 the Colonel should have a black holster, and belt like the M.P.s are wearing.
When J, K and Griffin enter MIB headquarters at the Brooklyn-Battery Tunnel ventilation building to fly to Cape Canaveral, the tilting shot prominently features three medallions over the entrance doors. Those cast aluminum reliefs, designed by Paul Manship, would not have been over the building entrance in 1969. They would have been mounted on the New York Coliseum convention center in midtown Manhattan. The reliefs were removed from the Coliseum in 1999 as it was being prepared for demolition and only later mounted on the ventilation building.
In 1969, we saw two versions of the neuralizer. The first was used at MIB when K was trying to nerualize J, rolling him into a gigantic version, but, later on, K had a portable one with battery power that he used on James. So why did they need the huge version if the portable version worked as well as it did?
At the stadium, when Griff is showing J and K the future game, right after he says the line "A miracle is what seems impossible...", in the shot from the back his hands are on their backs but on the shot from the front he is holding their hands by the elbows
When the 1969 cops have J pulled over, before pressing the button on the neuralizer, he is not wearing his glasses, but in the next shot, he has them on, despite the fact that he didn't have a free hand to put them on.
When J and K first encounter Boris on the rooftop of Wu's restaurant, Boris forces them off the roof, hanging on a door. When J and K hit the ground, there is a brief shot showing them falling in opposite directions. However, the very next shot shows them on the ground, side by side.
The bowl K is holding while telling J he is going to be suspended for 2 weeks, then 4 weeks, changes positions. K sets it down upright, then J says he quits and its upside down, then it's back upright.
In the "Oh, man. This coffee tastes like dirt." scene, Agent K says:
"How you know my name?" then picks up a cup of coffee with his right hand, passes it to his left hand and raises it to his mouth. The camera changes and K is seen holding the coffee cup in his right hand.
After J arrives in 1969 and leaves the Chrysler Building (42nd St and Lexington Ave), he looks across the street to a building labeled as the Roosevelt Hotel. The Roosevelt is at 45th and Madison Ave. In 1969, the Hotel Commodore was across Lexington from the Chrysler Building (the site now occupied by the Grand Hyatt).
When on the moon in the beginning, the Apollo lunar module is shown with the upper ascension module. This module was launched off of the moon to return the astronauts to the orbiting module to return to Earth.
J jumps from the roof of the Chrysler Building, which is 925 feet high. He falls for more than 30 seconds before activating the time jump device. Given the height of the building and the acceleration of gravity (32ft/s2), it should take less than 8 seconds to reach the ground (9 seconds if air resistance is taken into account).
The ARCNET device is shown activating shortly after the second stage separates, and the launch abort tower is jettisoned, which happens before Saturn V reaches even the low orbit. While at that altitude the sky would indeed appear black, as shown, the Earth behind the rocket is obviously too far away - it would appear like that from about half way to the Moon.
Saturn/Apollo stacks were taken to the launch pad with the launch tower preceding the rocket stack. They are shown with the Saturn/Apollo stack north of the launch tower instead of the other way around.
The uppermost swing arm of the launch tower is missing the "clean room" that allows access to the Apollo capsule before launch. It was normally moved away from the capsule several minutes before launch.
The motorcycle ridden by Boris the Animal has an low-profile rear wheel and tire which were not made in 1969. However, given that the motorcycle burns with a blue-green flame after it crashes, it is probably reasonable to assume it is at least partly of extraterrestrial manufacture.
The goof items below may give away important plot points.
In the climax of the film, J lets Boris shoot him three times so that he can memorize the pattern, go back in time, and dodge the shots. When Boris shoots the first time, there are three shots: one to J's left shoulder, one to his right, and a third to his left side. J even says the locations to himself, illustrating his desire to remember them. However, when J goes back in time and dodges the shots, Boris shoots a completely different pattern, including an extra shot: first he shoots to J's right, then two shots to his left, and a fourth back at J's right. Somehow J still calls and avoids all four shots.
After Agent J returned to the present on top of the Chrysler Building near to the end of the movie, the next shot seems to show the view of the Empire State Building from his point of view. In fact it's a view from the GE Building at Rockefeller Plaza.
In the first Men in Black, when K is deleting J's history; He deletes his birthday which was in 1975. In this one, James is shown as being about 4 or 5 in 1969. This would mean he was born no later than 1965, not 1975.
In the altered timelime where K is killed in 1969, Agent J's career should have been erased from existence because K recruited James Edwards from the NYPD to the MIB in Men in Black, but in the new timeline K would have been dead for 28 years before that happened. So who recruited James in the new timeline?
J used the time device to go back while fighting Boris at the end on the Apollo launch platform. When he went back in time there should have been two agent Js. Either that or when Boris went back in time he should have switched places with his younger self, and then there would be only one Boris.