Revealing mistake: When Frank is hanging off the roof, there is a pipe from which he tries to cling to. The first time we see it there is a white line where it will break. The next time we see it (just before it breaks) the line has been coloured in.
Revealing mistake: When Booth shoots the second hunter, you can see he that aims the gun at the mans stomach but the exit wound appears to be on the left side of his back, further up from where he aimed. (01:10:50)
Chosen answer: The two hunters had now seen his face so they had to be killed. If they reported Booth to the police as "a weirdo in the woods with a powerful gun" (or whatever) his face would become known to the authorities and his chances of getting anywhere near the president would be zero.
The_Iceman