What are the pros of Fourier-Galerkin spectral methods while solving PDEs?
Here’s the one that came in my mind first:
Easy implementation: using this method, differentiation operator computation is really simple since $$partial^phat{u}_k=(imath k)^phat{u}_k$$
Exponential convergence: let $u in C^m$ the exact solution, $u_N$ the numerical solution and $epsilon=||u-u_N||_p$. $$epsilonleq alpha N^{-m}||u^{(m)}(x)||$$
Therefore the convergence is exponential if $m=infty$.
With trigonometric basis functions your problem size is $N text{log}(N)$ instead of $N^2$.
Stabilization techniques are easy to implement and cheap:
Filtering in the modal space.
Zero padding in the modal space.
No aliasing due to the Galerkin ansatz.
Energy/Entropy stable disctretizations, e.g. via a skew symmetric implementation, are quite easy.
Cons:
You are restricted to periodic boundary conditions.
However, you may use a different basis, e.g. the Chebyshev expansion, with a fast DCT (Discrete cosine transform). This allows also computations with $N text{log}(N)$ and also Dirichlet or Neumann BC's.
However, then you are restricted to a Chebyshev grid.
You are restricted to smooth problems.
You are restricted to structured meshes.
But to be honest, it is hard to talk about pros and cons if you do not consider your application/problem. This should be the first step.