Do you remember the transmogrification device from Calvin & Hobbes? How about the many attempts at the transmutation of lead to gold, or the transfiguration magic of Harry Potter? Why was Jesus transfigured, and not transmogrified? Because these are extremely important distinctions, I went on a harrowing quest through the Oxford American Dictionary to find the differences.
Transmogrification is a real word, and it refers to a change of form of surprising or magical nature; if a gnome suddenly turns into a bipedal beetle with ape arms, that would be transmogrification. Its roots are somewhere in middle English. This is as opposed to transmutation, which is defined simply as "a change in form, nature, or substance," and seems to have a much more utilitarian usage. Transfiguration is a change into something more beautiful or elevated than its previous form, and is more appropriate for the art and spirituality realms.
I know there has been a lot of confusion about this all over the English-speaking world, so I propose an easy fix: all three words will be condensed into one, easy to spell and pronounce: Transmogrifigumutation. Or, transformation for short.