Full diffusion or Partial diffusion between s-box applications?
If the linear diffusion layer were applied twice without applying the s-boxes between them, it is possible there might be an iterative differential characteristic with only four active s-boxes per modified 'round' (i.e. instead of a provable minimum of 25 active s-boxes over 4 rounds there could be as few as 16 active s-boxes over 4 'rounds').

Consider for example a state where all four bytes in a single row are active.  After SubBytes, all four bytes are still active.  After ShiftRows, all four bytes in that same row are still active.  Then after MixColumns all sixteen bytes in the state are active.  Applying ShiftRows again doesn't change the fact that all sixteen bytes are active.  But applying MixColumns again might (assuming the bytes differences are just right) result in a single row of active bytes.  If that row is the same row as the one we started with, and if the resulting differences of the active bytes are the same as the differences we started with, then that is an iterative characteristic that only activates 4 s-boxes in each 'round'.

If you want complete diffusion in a single round while still retaining provable minimum numbers of active s-boxes, you should use a 16 x 16 MDS matrix with a branch number of 17, such as this one.  But even with such a beast, you can only prove a minimum number of 34 active s-boxes over 4 rounds.