Dylan's "Tomorrow is a Long Time" (from Greatest Hits Vol. II) vs. Nickel Creek's "Tomorrow is a Long Time" (from Why Should the Fire Die?).
The original versions of the above songs are obviously not terrible; Dylan sings much worse on many other records. But these particular covers of these particular songs are just better. Magnet's "Lay Lady Lay" more fully captures the sensuality (there, I said it) of the lyrics, while Sara Watkins of Nickel Creek has a certain breathless impatience in her voice that better serves the theme of "Tomorrow is a Long Time."
I note, however, that one Dylan song whose original version cannot be bested is Wigwam (from Self Portrait), which was recorded when Dylan was apparently in his feebly-moaning-to-horns period of songwriting. Not many people know this, but I too once went through a phase of feebly moaning to horns. It was when I was in a coma for a week, right after I had been in a car wreck. During that time, record company execs came into my hospital room and recorded my comatose moans and released them on iTunes without my family's permission. The result? You guessed it, my moans went platinum and the execs all became billionaires. True story!