She's not shot because it's more difficult to hit someone than TV and movies make it out to be, especially  when the target is moving around a lot, and the shooter is moving even more to keep from being hit.  Thankfully, there aren't many shooting schools for crooks.  It's not clear to me from that clip if he even fired a shot, because I can't seem to get audio on it.

It looks like she knocks the gun out of his hands.
Plus, I doubt he fired, because he was really in no danger at all, unless he was dumb enough to step closer to the counter.

I disagree with Todd, even someone who has never held a gun before can manage to hit a person they're standing across a countertop from.  He might miss once or twice if she was lucky, but if he was intending to shoot her, she'd be shot.  She's not shot because he obviously didn't intend to shoot anyone, just to rob the store.  I think the shock of someone swinging an axe at him, instead of cowering in fear of the gun, was disturbing enough that he probably never even thought about actually pulling the trigger.

It's also possible, even likely that the gun is fake or unloaded.  

Perhaps the crook watched "Raising Arizona" recently.

It doesn't seem like he ever fired.  So he probably didn't ever intend to use the gun.  Or this is fake.

I think the reason she doesn't get shot is that, while the robber has the better weapon, the store clerk is more ready to use her weapon.

In other words, the robber probably went in thinking "I have a gun, the clerk is a woman, all I (the robber) have to do is stick the gun in her face and she'll give me anything I want". When the robbery didn't go that way, and the robber realized he would have to shoot the women, he decided that killing her is a bad idea and that he'd better go to where the pickings are easier.

I think the guy just got freaked. Maybe he never intended to shoot (perhaps there weren't even bullets in the gun). He had planned on an easy robbery, and when the ax came out, he jetted..

Agreed with Todd H.  I've taught several people how to shoot, and you'd be genuinely surprised at how hard it can be for an inexperienced shooter to hit a 4 foot by six foot target from as little as 20 feet away.  Mass media would have you believe point and shoot is a guaranteed hit, but I've seen more than a handful of novices miss entirely under tightly controlled circumstances.  Female novices, oddly enough, are much better natural shots, though.

I'd like to know why he walks away from her at the very beginning of the video, giving her time to get the axe.

I would guess he walks away because the first approach was a "legitimate" transaction to get her to open the register to see if there's enough in the till to make it worth his effort and/or to gauge (inaccurately, it appears) what her reaction might be.  She may have gotten a bad vibe or an exchange of words or possibly saw his eyes pay too close attention to the register, so she moved for the axe ahead of the threat.  Of course we don't see the initial approach so this is pure speculation.

After watching the video, here is my take. When he was near the register, she saw that he was wearing a mask and cape. She quickly got her axe. Then, he walked up to the register to rob the store. From the moment that he walked up, she kept him off-guard by forcefully swinging her axe at him. He raised his gun, but when he moved in to shoot her or stop her defense, she swung the axe at his hands. He had to lower his gun. He raised the gun a second time. She swung the axe again, forcing him to lower the gun again and possibly injuring his hand. Then, his mask slipped and he had to adjust it. At that point, his window of opportunity was gone. A robber has to act quickly to avoid being caught, and this guy been in the store too long. He was also probably scared and bewildered. He gave up and fled the scene.

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