Each potential describes a different physical situation, i.e. they differ by the controlled variable of the experiment being described by them.
Take for example the Helmholtz free energy. Note that
$$
dF=-PdV-SdT
$$
so F is a function of volume and temperature: $F(T,V)$, since the equation reads that small variations in V or T causes a variation in F. It therefore describes a situation in which these variables are controlled, while their [conjugate variables][1] depend on them. An experiment with a heat bath confined to a constant volume can be described by this potential. A different experiment, e.g. when pressure is constant (and volume may vary), will be described by the Gibbs free energy, and so on.


  [1]: https://en.wikipedia.org/wiki/Conjugate_variables_(thermodynamics)