Laplace Transforms are a fairly common, if not required, topic in the first undergraduate course on differential equations. Essentially the Laplace Transform is used to treat problems with discontinuous forcing functions. It also has the advantage of naturally folding initial conditions into your solution. Typical problems allow for us to trade a differential equation in the time domain for an associated algebraic problem in the so-called frequency domain. The full mathematics is really analysis of distributions and you will not find much of that in the usual textbooks.
Fourier Transforms are not usually taught in the introductory DEQns course, at least in my experience at typical US universities. You might find Fourier analysis in a second course in DEqns where you focused on boundary value problems. However, I had such a course without the transform. The Fourier Transform would seem to enjoy greater popularity in its application to higher mathematics. Generalized Fourier analysis is a thing. Also, the application to quantum mechanics is important, the Fourier transform takes you from position to momentum representation.
Both transforms have a similar application, they're used to convert differential equations to algebraic equations. The Laplace transform treats initial data nicely. The Fourier transform implement boundary conditions naturally. As other comments have already pointed out, both of these transforms are best when implemented in tandem with the full suite of complex analysis tools. Last semester I taught an independent study course where the text (Introduction to Hilbert Spaces with applications by Debnath and Mikusinski) had an example which simultaneously used both Laplace and Fourier transforms in conjunction to solve a partial differential equation.