Difference between DFT and Z-Transform
Actually, the Z transform is not really a proper transform, just a re-interpretation of the sequence of samples as coefficients of a formal Laurent series.

In some cases the formal Laurent series converges, if it does, it does so on an annular region in the complex plane. For useful signals (stable, summable, exponentially decaying) this annulus contains the unit circle, and the evaluation of the Laurent series on the unit circle corresponds to the Fourier series.

The interesting point of connecting a signal sequence to a periodic function on the unit circle is the inverse transformation, that many useful sequences are sequences of Fourier coefficients. And of course that convolution of signals corresponds to point-wise multiplication of the functions.