This is a response to David's answer. Without having looked at that book yet I'd guess the problem is counting connected spanning subgraphs, because this is the point x=1 y=2 of the Tutte polynomial, and the author was interested in that. But in fact I think those three problems reduce quite easily from counting connected spanning subgraph problem. The following reductions should work for either exact counting or approximation, though I think the problem for approximation is still open.
Counting connected spanning subgraphs reduces to counting connected subgraphs (sketch):
Take a graph G in which we wish to count spanning subgraphs. Attach a $K_A$ to each vertex. If $A$ is chosen large enough, typical connected subgraphs of the resulting graph correspond N-to-1 to connected spanning subgraphs in G, where N is easy to compute.
Counting connected spanning subgraphs reduces to counting connected induced subgraphs (sketch):
Let G be a graph in which we wish to count spanning subgraphs. Divide each edge in two, so there are now |V|+|E| vertices. Attach a $K_A$ to each of the original vertices that were in G. If $A$ is chosen large enough, typical connected induced subgraphs of the resulting graph correspond N-to-1 to connected spanning subgraphs in G, where N is easy to compute.
Here's another interpretation of the question: what about counting unlabelled connected subgraphs? This is $\#P$ hard even for trees: L.A. Goldberg and M. Jerrum, Counting unlabelled subtrees of a tree is #P-Complete, LMS Journal of Computation and Mathematics, 3 (2000) 117-124.