Continuity mistake: Before crashing the limo into the fence, there are two bullet holes in the rear windscreen. Afterwards, there are five. Then from an inside shot as they start backing up, there are only two again.
Continuity mistake: Logan tries to give the Professor an injection, who pushes him away and falls over. As he falls back his left sleeve is still rolled up - the shot cuts and it's instantly right down to his wrist.
Answer: X-24's healing factor is much weaker and slower than Logan's, and Dr Rice has to give him the serum to heal his injuries at the farm, so presumably he couldn't heal fast enough to survive the shot. It may also have been a different bullet type - the one in Wolverine didn't do nearly as much damage on impact. Makes sense that Wolverine would have possibly designed one to overcome any healing factor.