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 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.