The early scenes, when the family first arrive in New York, are set in summertime, yet they drive through Midtown at night past people dressed in winter clothing, a Christmas-season Target billboard, and the Radio City Music Hall Christmas tree.
When Christy is helping Johnny learn his lines at the café (after visiting Sarah in the hospital), Ariel's two front teeth are adult teeth. In the next scene, when Ariel and Christy are in bed, they are back to being baby teeth.
As they arrive in Manhattan, a scene shows traffic congestion, the skyline, and the sky with a crescent moon. The crescent is facing the wrong way for an evening shot. It can't be dawn because a Dow Jones crawler reads, "Stocks finish mixed on Wall Street," and the scene has gotten darker, not lighter. Looks like they flipped the film unadvisedly. Phil Plait in his Bad Astronomy blog made a similar reversed moon observation about an episode of "The Simpsons."
When the Sullivans have Mateo over for dinner on Halloween night, they serve Colcannon: mashed potatoes with curly kale (or cabbage). Without a break in the action, the father leaves the table and the Colcannon has been replaced with bread.
In the shop in New York, a neon sign in the background reads "Take Away" which is actually the European phrase for take out food, and the sugar packets on the tables are tubes, which are also used mostly in Europe.
The camcorder would not be able to play back video in slow motion, as we see in the early frisbee scene. The shot was clearly made on film, overcranked, and then transferred to video to display on the LCD screen.
In the ball throwing scene at the carnival, as Johnny throws the last ball and the camera goes to a blurry slow-motion movement, you can see that the ball is still in his hand as it goes down yet a ball is seen entering the hole.