I think of how mpeg video cannot accurately portray looking at a rippling lake.. the sinusoidal motion does not have much in common with blocky mpeg.
The overtone movements become disassociated and start splitting up into disconnected sine wave blocks. Transients also become softened.
With MP3s this is obvious at the high end, while lows/mids sound passable. The compression algorithm has more frequency bins at the high end, so the damage is more obvious there.
There's also a loss of low-level detail. Reverb tails lose definition, and the music loses front-to-back depth.
That being said, age, tinnitus and amazing progress within the audio compression sciences have made it more difficult for me to separate uncompressed WAV and 320 kbps MP3, though the 320 kbps files in the article had noticable artifacts in most cases.