Is S-T CONNECTEDNESS #P-complete on instances when all s-t paths are of the same length?
After polynomial time preprocessing, you may assume that every edge in $E$ lies on some simple $s$-$t$-path.


If all simple paths between $s$ and $t$ are of the same length, then $G$ is series-parallel, with $s$ as left terminal and $t$ as right terminal vertex. This can be seen easily, since the forbidden subgraphs for series-parallel graphs (= subdivisions of diamonds) create two paths of different lengths between $s$ and $t$.
In a series-parallel graph, the number of spanning subgraphs containing a simple $s$-$t$-path can be determined in polynomial time by dynamic programming. 


In the dynamic program, we build up the graph by series and parallel compositions. For every resulting subgraph $G'$, we compute the number $p^+(G')$ of spanning subgraphs that do contain a simple path from its left terminal to its right terminal and the number $p^-(G')$ of spanning subgraphs that do not contain such a path. 


In a series composition on $G_1$ and $G_2$, we get $p^+(G)=p^+(G_1)\cdot p^+(G_2)$.
In a parallel composition on $G_1$ and $G_2$, we get $p^-(G)=p^-(G_1)\cdot p^-(G_2)$.
In both cases, the missing value $p^+(G)$ or $p^-(G)$ for $G=(V,E)$ can be computed from $p^+(G)+p^-(G)=2^{|E|}$.