Difference between the languages accepted by two DFAs with different initial state/accepting states?
Since $A,B$ are strongly connected, then if $q_1\neq q_2$, there exist words $p_1,p_2$ such that $\delta(q_1,p_1)=q_2$ and $\delta(q_2,p_2)=q_1$.

Consider case 2, then $w\in L(A)$ iff $p_2w\in L(B)$, and $x\in L(B)$ iff $p_1 x\in L(A)$. So you can add a prefix to switch between languages.

Consider case 3, then again - by strong connectivity there at most $|F_1|$ words $s_1,...,s_k$ such that for every $q_i\in F_1$ you have that $\delta(q_i,s_i)\in F_2$, and similarly for the other direction (from $B$ to $A$).

Thus, you can compose suffixes to switch between languages.

Combining these you can characterize the differences using prefixes and suffixes.
For example, in case 4, $w\in L(B)$ iff $p_1 w s_i$ in $L(A)$ for some $s_i$ in a predetermined finite set.

In fact, you can even say something interesting about these words: define $C$ to be the DFA where $q_1$ is the initial state and $q_2$ is the final state, then in case 2 you have $L(B)=L(C)\cdot L(A)$ (and similarly for the other direction).

As for the suffixes, things are more involved, since you cannot predetermine in which final state you will end. I'm not sure you can write this as a concatenation, but you can write $L(B)=\bigcup_{q\in F_1}L(A_q)\cdot L(E_q)$ where $A_q$ is the DFA obtained from $A$ be setting $F=\{q\}$, and $E_q$ is a DFA that starts in $q$ with final states $F_2$.

For case 4 you can combine the two.

You may be concerned that this is not a real answer, but rather just a characterization of properties using words rather than states, but this is a typical answer in this field (similarly to the Myhill-Nerode theorem).