I'm a physics graduate now working with computers. I study GR in my spare time to keep the material fresh. In the Wikipedia article about the mathematics of GR, one can read the following:
The term 'general covariance' was used in the early formulation of general relativity, but is now referred to by many as diffeomorphism covariance. Although diffeomorphism covariance is not the defining feature of general relativity, and controversies remain regarding its present status in GR, the invariance property of physical laws implied in the principle coupled with the fact that the theory is essentially geometrical in character (making use of geometries which are not Euclidean) suggested that general relativity be formulated using the language of tensors. [My italics.]
Do anyone know what kind of controversy the author(s) may be aiming at? Isn't general covariance, ehrm ... diffeomorphism covariance, a founding principle of GR?
Evidently there is no "right" answer to a question like this (unless you happen to be the author of said article and thus could share with the world what you where aminig at). Anyway, it seems as there isn't a widely known, heavily debated controversy regarding general covariance.
Even so, I've chosen to accept Ron's answer.
UPDATE 2: I've retracted the acceptance due to the linked article by prof. Norton. I think that, for all practical purposes, Ron's answer still stands, yet I want to review said article first. However, nobody should hold their breath waiting for me to figure this out. :)