Laplace transform is for exponentials, fourier is for the sine function (ie a complex exponential). So they do the same thing just with different weight functions.
All of this is just linear algebra, with different linear transforms. A fourier transform is a linear transform that transforms point space basis to harmonic oscillation basis.
All of this is just linear algebra, with different linear transforms. A fourier transform is a linear transform that transforms point space basis to harmonic oscillation basis.