Bit depth determines dynamic range. This means three things. First, we can capture a higher range of harmonics, or other sounds, coming from the sound source, that would, otherwise, be too quiet to capture. Secondly, our signal becomes louder. 1 bit approximates to 6 extra db of dynamic range. Thirdly, all things in nature (audio included) have a certani degree of noise. Higher dynamic range means that the noise floor, can be much lower than the sound source. For example, if you have a dynamic range of 6 db, then, if you capture a guitar but you have a noise floor of 3 db, you lose much of its clarity. However, if you have 96 db of dynamic range, it becomes much easier to get the guitar much louder than the noise floor.
A better example to demonstrate this is through digital pictures.
In the picture you see the immense difference between, 1 and 24 bits. As you add more bits, the picture gets clearer, but only in the 24-bit case, where we have 16 million colors we have a clear representation of the picture. The same thing goes with audio.