Suppose this three processes, at same T and P, each of them on thermodynamical equilibrium:
atoms -->molecule 1
atoms --> molecule 2
atomsdifferent --> molecule 3
Where atoms are infinitely separated, and may be different. The compositions of molecules 1 and 2 are the same, but molecule 3 has different atoms.
From experiments we can calculate thermodynamic functions. Suppose we have them.
Is there any thermodynamic function that able us to order those compounds in stability? Would you talk about G, about H?
Stability is only related to $\Delta G$, if $\Delta G$ all systems are stable. If Δ$\Delta G$ is different from zero, the bigger $\Delta G$, the bigger tendency to move from that conditions (more unstable). $\Delta H$ is not a criterion for stability.
It rests a question: why $\Delta G^0$ is different can't be interpreted in the same sense than $\Delta G$? From experiments we can calculate thermodynamic functions.