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: $F(T,V)$. It therefore describes a situation in which these variables are controlled. 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.
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