Laplace transform is for when you are going to do something professory or otherwise tricky, Fourier transform is for when you are going to do something engineery. Fast Fourier transform is for when you are going to do something so engineery it becomes tricky again.

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.

I'm currently in uni, learnt about both but has already forgotten what the Laplace transform is about, but still remember Fourier vividly as I got to play with it a lot.

If you read more into it's history, the DFT actually helped a lot during the nuclear arms race as it can detect any nuclear test that happened anywhere except underground.

Laplace is a generalisation of Fourier. Fourier can in principle only represent periodic signals. To circumvent this limitation, they invented the windowing of consecuive parts of signals and do the transform on each window. Laplace adds the transient part, that's what makes it more complicated.

That's not true. The Fourier series represents periodic functions. The Fourier transform applies to all real functions. They need not be periodic, although I'm not sure it's a particularly useful representation of aperiodic functions.

It's been a long time since I've thought about the Laplace transform, but IIRC, it has some properties that are preferable the Fourier transform as a math tool.

The Fourier transform better maps to an intuitive concept of frequency, which is why it is the basis for the transforms we actual use in practice in engineering, such as the discrete Fourier transform, which is what the famous FFT algorithm computes.