Helmet Pros and cons
PROS: The most significant benefit of mounting your GoPro to your helmet is that the view will be from the same point of view as you. The experience a watcher will have while watching your video will be very similar to what you experienced on that bike ride. If you look left, your camera will point left. That being said, if you tend to angle your helmet down and look forward to get a bit more of an aero position, your camera will be focused on the ground. Additionally, a helmet-mounted GoPro will also experience way less vibration than handlebar mounted. There are many dampers between the pavement and the camera itself.
Many have noted that the highest quality videos come from a helmet-mounted camera.
CONS: are that you're adding mass mounted off of your head. Your helmet will most likely need some adjustment to hold more firmly to your skull. In the event of a crash, there will be new things to break near your face, and you're modifying your helmet to do something it wasn't originally designed to do. Some have noted that the weight of the camera becomes noticeable on longer rides.
Chest Pros and Cons
This is probably the best setup for mountain bikers. Chest mounted camera setups will have the most stable camera angle. There will be minimal bouncing around, vibration, etc. If you're on single tracks with tons of ups and downs and obstacles, this is the best position. You will get a lot of footage of how the rider is maneuvering his or her bicycle.
That being said, your field of view is very restricted. If some action happens off to the side, you cannot easily focus on it with the camera. If you're already wearing a camelback or loose-fitting jersey, the mounting will most likely not bug you at all.
Handlebar Mount Pros and Cons
The handlebar mount is the most convenient and most comfortable to set up. Go pro mounts exist that integrate with a Garmin mount, so you never have to take the adaptor off your bike. The viewpoint is off the front of the bike, so if you have no interest in seeing hand position or what is below, then this is for you.
The biggest con for this setup is that of vibration and jerkiness. Handlebars are hung off the front of the bike, and any road imperfection can be amplified.
Also, it is important not to mount the GoPro directly to the bar (composite on composite) because it will transmit a lot of vibration. Try using a cut-up inner tube to act as a shock absorber by wrapping it tightly around the handlebars and then mounting your GoPro mount to that, clamping down tightly.