In explanations of the importance the tensors I often see people refer to transformation properties, general covariance and the like. Now, I have also often read that in principle any physical theory, e.g. classical mechanics and special relativity, can be written in a generally covariant form, but this would just make the formulation unnecessarily complicated. I'm now looking for a statement along the lines "CM has property X, which allows us to use vectors instead of the full tensor calculus." or "GR does not have property X, which necessitates the use of tensors." I assume "X" will have something to do with the no prior geometry feature of GR, but I'd like to make this more precise.