Are there strings which get accepted only by PDA by empty stack and not by PDA by final state, and vice versa?
Since it was not explicitly mentioned, I'll assume that the PDA in the question is a PDA $P_s$ that has been defined for acceptance by empty stack. Let the language accepted by $P_s$ be $L_s = N(P_s)$.

There definitely exists another PDA $P_f$ which produces a language $L_f = L_s$  by defining acceptance on final state. This is why we say that:


  A language L has a PDA that accepts it by final state if and only if L has a PDA that accepts it by empty stack.


However, it is wrong to expect that the two PDA $P_s$ and $P_f$ are the same.