I am trying to understand, to me, the difficult concepts of thermodynamic potentials, and I find their definitions numerous and hard to connect between each other. For instance, here are two definitions according to two different sources I consider quite credible:
Helmholtz free energy and Gibbs free energy according to Hyperphysics
According to Hyperphysics, the Helmholtz free energy is qualitatively described as:
The internal energy U might be thought of as the energy required to create a system in the absence of changes in temperature or volume. But if the system is created in an environment of temperature T, then some of the energy can be obtained by spontaneous heat transfer from the environment to the system. The amount of this spontaneous energy transfer is TS where S is the final entropy of the system. In that case, you don't have to put in as much energy. Note that if a more disordered (higher entropy) final state is created, less work is required to create the system. The Helmholtz free energy is then a measure of the amount of energy you have to put in to create a system once the spontaneous energy transfer to the sytem from the environment is accounted for.
And the Gibbs free energy as:
The internal energy U might be thought of as the energy required to create a system in the absence of changes in temperature or volume. But as discussed in defining enthalpy, an additional amount of work PV must be done if the system is created from a very small volume in order to "create room" for the system. As discussed in defining the Helmholtz free energy, an environment at constant temperature T will contribute an amount TS to the system, reducing the overall investment necessary for creating the system. This net energy contribution for a system created in environment temperature T from a negligible initial volume is the Gibbs free energy.
Helmholtz free energy and Gibbs free energy according to Wikipedia
However, according to Wikipedia, Helmholtz free energy
is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature and volume (isothermal, isochoric).
While, Gibbs free energy
is a thermodynamic potential that can be used to calculate the maximum of reversible work that may be performed by a thermodynamic system at a constant temperature and pressure (isothermal, isobaric).
I don't see how these two definitions connect with eachother, and my understanding of the two is now lacking. I definitely can see intuitively how Hyperphysics defines the two, but I don't understand the Wikipedia definition. I don't know what is meant by "useful" work other than the remaining energy that can be used after work must be done against ambient pressure to create 'space' for the system, and having a bit of energy in store due to ambient temperature providing some assistance raising it to a desired system temperature, but still, given the definition by Hyperphysics, it seems that the ideas behind enthalpy and Helmholtz free energy combined constitute Gibbs free energy yet according to Wikipedia's description Gibbs free energy and Helmholtz free energy seem like two sides of a coin rather than enthalpy and Helmholtz free energy being one.
Can someone provide an intuitive description of these potentials that can connect the two definitions I've provided? If there's a mathsy possible understanding that might be preferred, as I have more of a maths background. Also, I am unclear as to why these two potentials are so useful and prevalent. I know that a system is at equilibrium when $dG = 0$ which suggests it's an important parameter but I don't have a place for it in my head that's clear for its role in thermodynamics still. And I have less of a clue for Helmholtz free energy.