Taylor expansion is very useful when we approximate a complicated function near a point. Often we only have to know the first (or sometimes also second) derivative.
But in the case of expanding in a Fourier series, if the function is simple (like a polinomial) it gets more complicated.
If it is a complicated function, the integrals to calculate the coeficients may be very complicated (if analytic integrals exists).
It seems that it is the opposite of Taylor expansion: instead of simplifying, it complicates.
When is it useful to substitute an analytical function by its Fourier expansion?