In this paper, we address the enumeration of (induced) $s$-$t$ paths and minimal $s$-$t$ separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) $s$-$t$ hyperpath and minimal $s$-$t$ separator enumeration in a \emph{directed hypergraph}. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced $s$-$t$ hyperpaths and minimal $s$-$t$ separators unless $P = NP$, and if there is an output-polynomial time algorithm for the $s$-$t$ hyperpath enumeration, then the minimal transversal enumeration can be solved in output polynomial time even if a directed hypergraph is $BF$-hypergraph. Since the existence of an output-polynomial time algorithm for the minimal transversal enumeration has remained an open problem for over 45 years, it indicates that the $s$-$t$ hyperpath enumeration for a $BF$-hypergraph is not an easy problem. As a positive result, the $s$-$t$ hyperpath enumeration for a $B$-hypergraph can be solved in polynomial delay by backtracking.