$\lVert A - B \rVert_F^2 = \lVert A\rVert_F^2 + \lVert B\rVert_F^2 + 2\langle A,B \rangle_F$. We know that $\text{min}\lVert A - B \rVert_F^2$ occurs when $A=B$ which implies $\lVert A\rVert_F - \lVert B\rVert_F= 0$.
Let $A\ne B$, can we obtain $\text{min}\lVert A - B \rVert_F^2$ possibly in terms of norms $\lVert A\rVert_F$ and $\lVert B\rVert_F$? Is this feasible?