What is the advantage of music with more dynamic range?
If you compare two short passages of a song (or of two similar songs), where one passage is louder than the other, then the louder passage will at first sound "punchier." The reason is the nature of our sense of hearing, which can be explained by the Fletcher-Munson equal-loudness curves. Read more on this in the section Why do strongly compressed tracks seem more attractive?
The advantages in sound quality of more dynamic music becomes apparent when the two passages to be compared are heard at the same volume (sound pressure) level. It is important to understand that equal loudness means the same loudness as you perceive it, not according to the volume knob on your stereo system. The longer you listen to the song, the easier it is to appreciate the difference. As a result of natural sound characteristics, dynamic music does not cause headaches and does not make people more aggressive. It does, however, faithfully reproduce dynamic events in the music as such. On the CD master of "Californication" (Red Hot Chili Peppers), there is no perceptible change in dynamics. But the unmastered vinyl version has dynamic increases of up to 3dB within a single song. Similarly, a crescend (shorter increase of loudness) is flattened out and practically imperceptible on over-compressed masters, whereas these passages obtain their full range of musical expression when the sound is more dynamic.
This video by Matt Mayfield demonstrates how important this is:
From a more objective standpoint, over-compressed music sounds flat and weak.