When Booker first goes to see the horse at the hotel and it runs out of the barn it has a big white blaze that covers most of the face, but in subsequent shots it has only a small star on the forehead.
When Grace is watching the show jumping, the commentator says "Pilgrim moves with absolute confidence at the command of Ms. McLean." This makes it clear that it is Grace and Pilgrim jumping. However, the Pilgrim on the TV is a bay with a white blaze. The Pilgrim throughout the film is chestnut with a small white star.
There is inconsistency with the mud covering Pilgrim throughout the movie. Even though he has been almost submerged in water on the ranch at least twice, the mud covering him is present, then it disappears, and is back again, as if the horse never was exposed to any water at all.
Tom Booker removes the rope looped around Pilgrim's pastern, but when the horse is getting to his feet after being "rocked up" the rope is still looped. When Grace begins to ride him, the rope is once again off.
Briefly, we see the legs of Pilgrim sliding down the bank ahead of the other horse but it is the other horse who reaches the road first. Also Pilgrim alternates between standing and lying as he slides down the bank.
When Grace first wakes up in the hospital there is a close-up of her face and a cloth or bandage can be seen on her forehead. On the revealing shot, there is nothing there and Annie is seen grabbing a cool cloth and placing it on Grace's forehead.
In an early scene, a phone call is placed from Manhattan to the suburbs. The audible touch-tone sequence is seven digits. Since the NYC metro area has many area codes, ten-digit dialing is mandatory there.
Booker is in the corral alone trying to "hook on" with Pilgrim. Just before he squats to remove the lasso from Pilgrim if you look under the horses neck you can see the lower half of a man dressed exactly the same as Booker.