The path between any two nodes in cyclic directed graph
If the graph happens to be a DAG, then you can found the number of paths from any vertex to any other vertex using linear algebra. Let $A$ be the adjacency matrix of the graph. Compute
$$ B = \sum_{k=0}^{n-1} A^k. $$
Then $B_{ij}$ is the number of paths (of length at most $n-1$) between vertex $i$ and vertex $j$. This also works for weighted graphs, in which case the weight of each path is the product of the weights on the edges.

It is easy to construct examples where there are exponentially many paths. For example, let the vertices be $x_0,\ldots,x_n,y_1,\ldots,y_n,z_1,\ldots,z_n$, and the edges be $(x_i,y_{i+1}),(x_i,z_{i+1}),(y_i,x_i),(z_i,x_i)$. The number of paths from $x_0$ to $x_n$ is $2^n$.

If the graph is not a DAG, then there can be infinitely many paths from a vertex to another, since any given cycle can be taken an unbounded number of times. In this case it makes sense to ask for the number of simple paths from one vertex to another. In the weighted version (with weights of polynomial length), we expect this to be difficult, since HAM-PATH can be solved this way.