They are indeed related, which shows up in many places.
The difference is in the meaning of the result, but most importantly, in the domain. Fourier transform requires a $L^2$ integrable function (physicists abuse this a lot). It gives you a spectrum of the signal, it treats positive and negative times equally (noncausal signals). It's a unitary transformation (it's its own inverse, plus or minus a conjugation and a sign) and as such has a lot of nice mathematical qualities.
Laplace transform deals with causal signals: typical examples are response functions in circuitry, material physics and so on. The lower integration limit is 0 (or finite, but not $-\infty$). It is more related to a fourier transform of a function, multiplied by the Heaviside step function. The difference is also in what values you usually put in. Although you can put complex values in both of them, the traditional domain are real frequencies for FT and real (or at least positive complex) exponents for LT. But their relationship does show up in physics a lot: fluctuation-dissipation theorem, relationship between statistical physics and quantum mechanics, one of the prominent examples is the the dielectric response (refraction index) as a function of frequency. As FT, it's also suitable for solving linear differential equations, but has the advantage, that its causal nature takes the initial conditions into account explicitly and ignores the negative time portion. That's how most phenomena in nature work (future does not affect the past, nothing happens before the cause happens). FT on the other hand deals mostly with uniform steady-state phenomena that have been running since forever following the same rules. Periodic phenomena are one big example of this.
Z-transform is, as you noticed, discrete. It is quite related to the discrete Fourier transform, zhe $z=1$ circle representing the fourier spectrum. It's a sort of polar version of the fourier transform and makes the cyclic nature of frequencies in discrete signals apparent (compare to the FT where the frequency domain is infinite). The interpretation of the $|z|=r$ coordinate gives you the dissipation rate, making it suitable for real time series analysis (stability of linear systems and so on).
In short: all three transforms follow the same concepts, but the difference is both in what kind of signals you put in, and what sort of information you need to get out of it.