The electromagnetic potentials ##\Phi## and ##\vec{A}## are not uniquely defined by the physical situation and thus are not observable fields in Maxwell's theory. The electromagnetic field ##(\vec{E},\vec{B})## are observables.
The potentials are introduced as "auxiliary" mathematical quantities to fulfill the homogeneous Maxwell equations identically, i.e., you set
$$\vec{B}=\vec{\nabla} \times \vec{A}, \quad \vec{E}=-\partial_t \vec{A}/c -\vec{\nabla} \Phi.$$
Then indeed the two homogeneous Maxwell equations,
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}+\partial_t \vec{B}/c=0$$
are identically fulfilled (check this!).
But for given ##\vec{E}## and ##\vec{B}## also
$$\Phi'=\Phi+\partial_t \chi/c, \quad \vec{A}'=\vec{A}-\vec{\nabla} \chi$$
lead to the same ##\vec{E}## and ##\vec{B}## as ##\Phi## and ##\vec{A}## for any arbitrary field ##\chi## (check this!). That's the famous gauge invariance of Maxwell's electromagnetism.