What Is the Fourier Transform?

They are quite related, but the Fourier transform seems far more beautiful and generalizable: you can do 2-d, 3-d, etc transforms, and they automatically respect the symmetries of the problems (e.g. rotating the coordinate system rotates the Fourier transform in a corresponding way; frequencies and wave-vectors have meanings). This fully extends to any "nice" abelian group satisfying minor technical conditions, where the mapping is to it's dual group. It even mostly extends to non-abelian groups (representation theory), though some nice properties are lost.

The Laplace transform shines in having nicer convergence properties in some specific cases. While those are extremely valuable for control problems, it really is a much more specialized theory, not nearly as widely applicable. (You can come up with n-d versions. The obvious thing to do is copy the Fourier case and iteratively Laplace transform on each coordinate; the special role of one direction either directly in the unilateral case, or indirectly via growth properties in the bilateral case make it hard to argue that this can develop to something more unifying; the domain isn't preserved under rotation.)