At he beginning of the horse-breaking scene the horse's hair is a mess and covered in mud. Once the horse is lassoed, the horse is suddenly clean and brushed. At the conclusion of the scene, as it runs away, the horse is muddy again.
In the opening sequence, the wording says, "In 1901 he drifted into Wyoming 'Territory'". Wyoming had been a state since 1890.
The rifle Steve McQueen uses in the film is a Winchester Model 1876, stated to be chambered in .45-60 Winchester, however, the rifle was actually chambered in .45-75 Winchester. The real life Tom Horn carried a Winchester Model 1894 chambered in .30-30 Winchester.
If you're trying to sneak up on someone, it's generally a good idea to take your spurs off first.