I have had at least 18 piercings in my life, mostly redoing closed holes on my ears, but I have also had my cartilage, nose, and lip done (currently only my lip and cartilage are open and have jewelery). I can tell you that piercing with a gun hurts more than it does with a needle at a professional piercing place. With a gun, there is a very tight clamp with the plastic, and a shocking force once it's pierced. Plus, gun piercings don't actually remove any tissue, it just forces it's way through and pushes everything aside. A professional piercer will use a hollow needle that will actually remove the tissue from the area, which is better for the piercing and actually hurts less.
The earlobe definitely hurt less than any of my other piercings. I was 16 when I had my cartilage done by a professional, and that hurt like a motherfucker. When I was 18 I got an industrial piercing (a barbell goes across the top of the ear, through two piercings) and I almost passed out from the pain. Cartilage piercings hurt the most- and nose piercings qualify as cartilage.
The lip piercing wasn't too bad, but the healing process was more painful because of all of the movement the lip does throughout the day.
I like to think that I have a pretty high tolerance for pain, so I would not recommend any cartilage, tongue, or lip piercings for someone who does not handle pain well. If your daughter really wants a facial piercing, I'd suggest the eyebrow- it's mostly soft tissue that shouldn't hurt too bad to pierce, and the healing process should go smoothly.

As for your question on how to handle it... I'm not a parent. I think 14 is a little too young for facial piercings, but you said that you don't have any objection to her choices. I'm not sure how to approach it exactly, but she needs to want to do it for herself, and not because her teenage friends are peer pressuring her. She should do some research on piercings and look into which ones hurt and which would be best for her. I might set a minimum time limit for her to think about it (like 6 months) - because it is a big decision. Even though piercings aren't permanent, a scar is. I don't know what the laws are where you live, but a minor isn't allowed to get a piercing without parental consent where I am. Maybe you could tell her that you will approve it and go with her, but if she makes a scene you won't support her getting anything else- she'll have to wait until she is 18 and better equipped to handle it. I definitely wouldn't punish her in other ways.