Towards the beginning of the film, Rhoda's Google searches "John Burroughs and new haven and accident", but when the results are displayed, the search box (which displays the terms which have just been searched) reads just "John Burroughs".
When John Burroughs is taking his pills after working on the outside of his house, the glass is 1/4 full, on the next scene when he drinks the water, the glass is 3/4 full.
John's address as shown on his website is 12 Russel Road, however a later scene shows his address on his personal check as 1 Russel Road.
After the car accident, we see a hole in the windscreen and John's son laying a few meters away from the car wreck, suggesting the impact had caused him to be thrown through the windscreen onto the street.
In a flashback later in the film, we can clearly see that John's son is wearing a seat belt immediately prior to impact, which would have prevented him from being thrown through the windscreen as a result of the accident.
The web page Rhoda clicks in her search is "www.nhc.com" but the layout of the page she pulls up is clearly that of the NY Times (when the movie was made, and the current layout as of Nov. 2011).
Rhoda tells the story of the first man in space, who was supposedly in orbit for almost a month. In reality, Yuri Gagarin's Vostok 1 mission lasted about 108 minutes from launch to landing, with a little over an hour (a single circumnavigation) spent in orbit. Only she uses this example to tell a metaphorical story, so the narrative part of her statements does not have to be based on reality.
When John asks Rhoda if she wants to look in the telescope at Earth II, the lens cap appears to be on the end of the telescope. However, the "cap" is actually a aperture stop, that is a cap with a hole in it, which becomes clear momentarily near the end of the scene just before she quips, "I wonder if I am cleaning your house (on Earth II)?"
At 17:20, Rhoda opens her laptop to look at an internet contest to win a flight to earth 2. However, the address bar in the browser points to file:// , so it shows a local file from her own laptop.
When Rhoda 'googles' Burroughs, the text of the main article about him repeats word-for-word in a second article, just below Burroughs' picture on that same page.
Around 1:12:00 after Rhoda and John toast there is a shot of the house from the outside with Earth 2 at "Full Earth" - the entire face is lit. A second later a shot from inside the house of Earth 2 through a window shows a partially-full Earth 2. Earth 2 is not close enough for the apparent angle to cause a phase shift of this magnitude; Earth 2 still would have been a Full Earth when shown from inside the house.
When Rhoda runs out of the house after the television broadcast, the cameraman's shadow is visible on her for a few seconds.
The movie takes place in Connecticut, but when Rhoda is walking down the street and is approached by the man wearing a foil hat/sandwich board, the Los Angeles skyline is clearly visible.
If a planet the size and density of Earth appeared so close to itself, the gravitational pull would cause significant changes in its orbit and 'Earth One' would suffer from constant disasters, e.g. floods, earthquakes, thunderstorms.