My understanding is that the crux is that in exchange for a very simple and easy licensing of the original, Jonathan Coulton signed away all of his rights except for his rights to his actual performance/recording.
Weird Al gets more complicated (and presumably expensive) licenses of the originals, so that he full out owns his creations.
That can't always be true, because in the debacle over Amish/Gangstas Paradise he stated that he was misinformed that the original artist has OKed it as a matter of courtesy, but that he had parody freedoms anyway.
Weird Al gets more complicated (and presumably expensive) licenses of the originals, so that he full out owns his creations.