A friend was once praising a particular cover of "Hallelujah" by someone on one of those Star Search sort of television shows. The singer was maybe 20. While the singer was quite good, the song's power is in the lyrics as well as the music, and it is a rare 20 year old who can deliver this song with an understanding of its deeply dark undertones. The link above is tinny, so you will just have to trust me when I say Rufus Wainwright and Leonard Cohen are equally up to the emotional part regardless of how good their voices are. You can feel the twist in their guts as they each deliver the song.
Another performance that exemplifies what I'm trying to get at is Kurt Cobain's version of "Where Did You Sleep Last Night". While I'm fairly certain he knew nothing of losing a spouse under the wheels of a train, you can feel that he knows what it means to be "in the pines". At the outset, you know the song is ominous and my breath freezes in my chest as he moves through the chorus at the end.
Whatever that quality is, Fiona Apple has had it since she was seemingly too young to know what she was talking about. So I know it's not all age and experience.
And whatever it is, I think it's why Bob Dylan's music works- I mean, it just cannot be the quality of his voice exactly.
While I can enjoy a light pop song as much as the next person, I will steadily choose those that seem to know what they're talking about and mean it, whether or not I can grasp their meaning.