Well, if you assume all 12,500 people are of the same ability as the one person (perhaps it is sufficient for their abilities to be randomly drawn from the same population distribution, but I'm not sure of this), then you're taking independent observations from the same distribution, and both situations are essentially identical. However, you'd probably learn something after a couple thousand shots, so I'd give the 1 person a better chance.

The odds can vary a bit on where the hole is located and what kind of grass is next to the hole. When you are looking at these averages they are hitting a golf ball on to a green, which typically is rather soft and allows the ball to stop relatively close to where it lands. A well hit shot that lands on the green with backspin, will slow the ball down quickly, and then it rolls at a speed which may be slow enough where it can fall into the hole.

If you are trying to hit a ball into a hole at a driving range, the driving range could have either longer grass that slows the ball down quicker or alternatively, it could be on concrete with artificial turf, both of which would make it harder for someone like a PGA professional to find the optimum place to hit the ball.

That being said, it would have been an interesting business prospect to try an arbitrage the contest by buying a large amount of tickets and hiring a PGA pro.