A pixel in an 8-bit grayscale image carries one of 256 possible values. That's eight binary digits — eight bits — used to encode intensity. The smallest detectable difference between two pixels is one of those 256 steps: about 0.4% of the full range.
A pixel in a 16-bit image carries one of 65,536 possible values. The smallest detectable difference is 0.0015% of the full range — 256 times finer. For samples where the bright signal is, say, 100 times stronger than the dim signal you care about, 8-bit may not even resolve the dim signal as a distinct value from the noise floor; 16-bit will. This is why scientific cameras almost always acquire at 16-bit, and why downcasting at the start of a pipeline is rarely a good idea.
The flip side: 16-bit images take twice the disk space, and most casual viewers display them by linearly compressing to 8-bit anyway. So the gain is purely quantitative — it shows up in measurements, not in what the eye sees in a quick preview.