Fractional polynomials and cubic splines each have the defects of their virtues.
The point of cubic splines is to be local and flexible and smooth and able to approximate any smooth curve.
The point of fractional polynomials is to not be able to approximate arbitrary smooth curves easily, on the belief that arbitrary smooth curves are not a statistically relevant class of functions. They are not local and much less flexible, again on the assumption that too much flexibility and localness is bad.
In settings where it's sensible to believe in fractional polynomials, they will be superior because they don't have unnecessary flexibility and they are sensitive to data across the whole range of $x$ in fitting $f(x_0)$. In settings where it's not sensible to believe in them, they will be inferior because they don't have necessary flexibility and they are sensitive to data across the whole range of $x$ in estimating $f(x_0)$.
You're unlikely to get agreement about which settings are which, but as an example:
- modelling the relationship between wind and air pollution, I might be happy to use a fractional polynomial (perhaps after log transformation), because there could be fairly simple physical relationships to mixing volume of the atmosphere
- modelling the relationship between time of year and air pollution, I would want some generic smoother such as a regression spline, because I don't think there's a simple relationship of any sort.