Hawaii had only become a state about 8 years before this movie was made. Perhaps no one knew how to properly pronounce some of the more obscure Hawaiian names, but when Roy was orbiting over Hawaii, Fred Gifford (Leslie Nielsen) says he is now over the "Kauai tracking station", pronouncing it "Cow-Eye".
The NASA weather reporter indicates sky conditions will be "CAVU" (clear and visibility unlimited) for the launch. When Roy arrives at the space center on launch day,though, it's an overcast sky. Just minutes later, at the base of the rocket, it's mostly sunny with a few clouds.
When Fleming respools the capsule's programming tape, he has accidentally gotten the peanut butter mixed in with it. The tape reels are so clogged, they cannot play. Later, when Fleming looks out the capsule window at Hawaii, the tape reels are seen in the background running normally. The peanut butter then returns in subsequent shots.
When in the space capsule, the strings holding the microphone, the doll, the crackers and peanut butter are visible. When Don Knotts is floating around the capsule, the strings holding him are also visible.
When where Roy (Don Knotts) accidentally opens the door to the centrifuge and is subject to high winds pushing him down the hall, you can clearly see 2 shirts and a pair of boxer shorts are attached to a single wire and quickly pulled down the hall.
When Roy is in space, he's shown doing somersaults while weightless. One of NASA's ground personnel who is monitoring him reports that "he's rotating at a rate of 360 degrees per second", which means that he'd be making a complete rotation every second. His true speed of rotation is nowhere near that fast.
When Roy sees his Pa and friends from back home coming to visit the space center, he leaves the buffer and it's still running on its own. Buffers have a 'dead-man's switch' that turns it off when you aren't holding it.
When Roy is giving his father and his friends a tour of the space center, he becomes trapped in a capsule. While looking for a handle to help extricate Roy, his father brushes his arm against the capsule window, damaging it (it appears to be made out of crepe paper at this point). Immediately after, Roy punches the window out from the inside; a freeze-frame shows the damage is gone just before the window breaks.
In the first airport scene, the airliner's right-hand engine is started with no apparent effect on the passengers standing behind it. In reality, they would have been enveloped in a cloud of blowing smoke and dust, and loose articles such as the men's hats would have blown away.
When Roy arrives at the Johnson Space Center in Houston, he goes to the personnel office, but it has a sign out front that says it's closed from 12PM until 2PM, and it's obviously between those times as he goes to the cafeteria to wait. But his shadow is very long, showing the sun is overhead at about a 60-degree angle, or sometime in the early evening, not the afternoon.