Time-invariant continuous model:
$\dot{x}(t) = Ax(t)+Bu(t)$
$y(t) = Cx(t)+Du(t)$
Time-invariant discrete model:
$x_{k+1} = Ax_{k}+Bu_{k}$
$y_{k} = Cx_{k}+Du_{k}$
Why does the continuous model result in a rate of change $\dot{x}$, while the discrete model results in a new state $x_{k+1}$?
It is not how the discretization is normally done, but see what happens if you replace the derivative by the forward difference for small $h>0$ $$ Ax_k+Bu_k=\dot x(kh)\approx\frac{x(kh+h)-x(kh)}{h}=\frac{x_{k+1}-x_k}{h}. $$ Thus, the rate of change is approximately proportional to the discrete state difference, and one can write $$ x_{k+1}\approx (Ah+I)x_k+Bhu_k. $$
