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