Why do we use dB to represent the difference between two voltages?
Many processes in nature are either of logarithmic nature (like human senses) or have a great dynamic range.

Describing them on a logarithmic scale and expressing differences in dB has several advantages:


often the absolute difference doesn't matter, but the ratio (that's what dB is used for) does (e.g. signal-to-noise ratio)
smaller numbers can be used
there's an approximately linear relation between measurement and perceived sensation
chained attenuations or amplifications can be expressed by addition instead of multiplication (easier to calculate in the head)


Here's another video about it.