Why do dynamic songs sound better than over-compressed songs on the radio?
On the radio, CD masters undergo a great deal of processing. The more compressed a track is, the more this processing damages the sound.
The first processing step is data compression (or data reduction) of the original CD files (Wav-LPCM) in order to archive the material on the radio station server. In the section Why do data compression processes (mp3, for example) strongly affect loud masters with no headroom? you can read more about why loud masters lead to significant distortion and degradation of sound quality even before the hacked up audio material gets to be broadcast, when it is submitted to further processing.
This processing chain consists of very modern DSP (Digital Signal Processing) algorithms which give each radio station its own sound, but also make everything equally loud.
If you have a song with DR14 (14dB of useful dynamic) and then make a DR6 version of the same song to be sent through this kind of processing chain for radio broadcasting, then both versions will be perceived to be of equal loudness. The only (fundamental) difference is that the over-compressed DR6 song is further deformed and loses the integrity of the loudness relationships, while the DR14 song survives the broadcasting chain relatively intact and is therefore much more agreeable for the ears.
These facts have been carefully confirmed and scientifically proven.