The mathematical formula of the Laplace and Z transforms are same with just one difference. I.e. in the first we use $t$ for continuous-time signal and in the latter uses $n$ for discrete-time signal. I don't think that there are any other differences.
While discussing Fourier transform for continuous-time (normally analog) signals, we use the continuous Fourier transform and for discrete-time (most often digital) signals we use the discrete-time Fourier transform (DTFT).
So my question is why the Z transform is considered a separate transform? Why it is not named the "Discrete Laplace Transform" ?