Both fourier transform and z transform can convert discrete time domain to frequency spectrum domain. Then why do we use fourier transform rather than z transform? What is the reason behind it? Both give us the frequency spectrum we want.
Then what about Discrete Fourier Transform? How is it different from z transform?
The purpose of all transforms is to reduce the mathematical description to a simpler algebraic expression.
Use of the Laplace transform in the s plane representation for frequency analysis of sampled (digital or discrete) data is made difficult by the the need for infinite polynomials with infinite numbers of poles/zeros.
The Laplace transform is equivalent to the (continuous) Fourier transform.
(It is identical to the one-sided Fourier transform with just a different choice of frequency variable.)
Both are badly suited for discrete signals, because, as you say, they yield expressions that are hard to manage then.