The Laplace domain is otherwise known as the s-domain, where s = σ + jω (having both a real and imaginary part). Laplace transform is useful for converting differential equations to an algebraic equation which is much easier to solve. The Fourier domain is widely known as the frequency (ω) domain - just a function of jω, no real part σ. This is handy in electrical engineering - signals/systems, communications, etc. - because FT allows you to see the frequency components of a time signal. The math is very similar; they both use an integral - LT contains an exp(-st) in the integral where FT contains an exp(-2pi*jωt). So you can see the similarities which are also visible if you ever compare transform pair tables for FT and LT. The basic difference is σ = 0 for FT. The frequency variable is easy to grasp - all time signals can be broken down into sinusoids of different frequency. But the variable s is purely mathematical. Either can be used in many cases (personal preference); they differ in certain properties and assumptions, but are quite similar in use and actual conversion/math.
Trying to get a good grasp of certain underlying mathematical techniques (Laplace Transform) in engineering can be challenging/frustrating. I hope i didn't lose you in my rambling. But those are my thoughts on the differences b/w LT and FT from several years of EE in college - while it's still fresh. The FT is quite interesting and very useful in EE.