Higher-end productions will always shoot at the highest bitrate (or data rate) the camera offers. As simple as this is, higher bitrate means higher quality, but also eats up more space. For DVD-quality (720x480), it's about 4-8 Mbps, while Blu-Rays are at least 25, but can be a lot more what with UHD 4K around and 8K a few years down the road.
The reason both settings are offered, simply put, are to determine whether you want quantity or quality. If you're shooting for fun but want to take a lot of video, you might choose 10.4, but if you're shooting a project or feature film, you'll want to go with 15.7--optimally you'd have an on-site editor who can dump the SD card (or hard drive) to a laptop, tablet or desktop workstation and clear it for you, since 15.7 will eat through your available space quickly. Or just have a lot of cards on-hand in these situations.
As far as technically what additional information is stored, it really does depend on the manufacturer, the camera model, and the codec they are using for recording--15.7 stores more data, period--you may see more well-defined hues, sharper edges, less chance of "blocking," truer skin tone, and just more detail in general. This is all stored as ones and zeroes, so the exact specification on what it's recording is simply that it's recording enough additional detail to take up about 33% more drive space.