Calculating discrete-time transfer function
My Z table doesn't have 1/s3.  It has 2/s3.  The Z transform for 2/s3 is;

\begin{equation} {T{^2}z(z+1)\over (z-1)^3} \end{equation}

To make this work with 1/s3, you have to take multiply this transform by 1/2. Using this and your equation;

\begin{equation} {(z-1)/z} \end{equation} 

gives the Matlab result.

Here is a link to a second table that confirms the transform I have;
Laplace to Z transform table with 1/s^3