Two inequivalent gauge transformations of $\mathbb{A}_\mu=0$, described by $U$ and $\tilde{U}$ of a pure $SU(N)$ Yang-Mills theory as $$\mathbb{A}_\mu=\frac{i}{g} U\partial_\mu U^\dagger~\text{and}~\tilde{\mathbb{A}}_\mu=\frac{i}{g} \tilde{U}\partial_\mu \tilde{U}^\dagger\tag{1}$$ represent two different minimum energy classical field configurations. They can be inequivalent. Hence, it is possible to have more than one minima of the Hamiltonian.
In $U(1)$ electrodynamics, we can have two different gauge transformations of $A_\mu=0$, $$A_\mu=\frac{i}{e} \partial_\mu\theta(x)~\text{and}~\tilde{A}_\mu=\frac{i}{e}\partial_\mu \tilde{\theta}(x)\tag{2}$$
Question In what sense, $A_\mu$ and $\tilde{A}_\mu$ of (1) are equivalent but $\mathbb{A}_\mu$ and $\tilde{\mathbb{A}}_\mu$ of (2) are not? Mathematically, what does it imply to say that $A_\mu$ and $\tilde{A}_\mu$ are not different?
