Everyone panics when Barney gets trapped in the bank vault, yet in the season 2 episode "Crime-Free Mayberry," Andy revealed that everyone is aware that a door has been cut into the wall because the bank lost the combination and the company that built the vault went out of business.
After Barney breaks through the wall, he starts walking toward the door while taking his sweater off. Once he arrives outside, his sweater is back on and he begins to remove it again.
Andy and Barney talk about the movie "G-Men" and state that Glenn Ford was the star. Actually, the 1935 Warner Bros film "G-Men" starred James Cagney, not Glenn Ford.
When Barney foolishly closed the bank vault door with him inside, anyone could have easily pulled opened the vault door since it does not lock when only closed. Bank vault doors lock when you turn a large wheel to engage the pins and the time lock if it has one.
After Andy enters the bank, the real bank robber tells Andy to toss his gun down. Andy showed up thinking it was Barney pulling another fake robbery, so he wouldn't have brought along a gun. Andy seldom carries a gun, even when he is facing a real threat and did not bring a gun with him. The gun that the robber tells Andy to drop was taken from the robber's surprised partner, as Andy came in the door.
Fairly late in the show, Barney writes the getaway driver an extensive ticket. Behind him, across the street, is Floyd's barber shop, the TV & repair shop and the grocery store. For the latter, the shelves in the display window have no groceries and there are no groceries on the sidewalk. Also, there are no product signs in the windows. It also appears that the barber shop is without a name in its window, too.
The bank is shown to have a large writing table upon which customers can write out deposit slips/withdrawal slips. The table is still there at the beginning of the actual robbery. When Sheriff Taylor and Henrietta (the bank employee) enter, the writing table has disappeared, apparently for the sake of the hostage-taking action scene.
When Andy says, "You got the rundown, Glenn?," the tone of his voice changes, revealing it's a voiceover (presumably to cover a production problem).
When the bank robber tries to fire Asa's Colt Peacemaker, the strings that are used to pull away the parts when the gun falls apart are visible.
At time stamp 20:53, the bank robber's gun falls to pieces. The strings going out from each side, which create the stunt, are visible.
Barney breaks through the wall and into the beauty shop. At least two goofs exist, possibly three: #1) A bank vault would not have an unarmored wall adjacent to the building next door. #2) The bank is clearly made of brick and/or stone and the wall would be impossible to break through even if the vault did have an unarmored wall. #3) The bank and beauty shop are two different buildings and likely do not have a common wall, much less one made of only lath and plaster.