Max cut problem between two connected subgraphs
Here is a straightforward reduction from the max-cut problem: 

Take any graph and add two new vertices $u,v$ and connect them to every other vertex with weight 0 and connect them to each other by a very very big weight edge. In an optimal solution to the above problem $u$ is in one partition and $v$ in the other (this also ensures that each part is a connected component), hence an algorithm that solves this problem in the new graph, could solve the max-cut problem in the original graph and vice versa.

P.S: I supposed we can have edge weights as otherwise a similar argument works however instead of adding vertices we add two long paths (each of length $n^2$), for $i\le n$ we connect the $i$'th vertex of the path $P_j$ (for $j\in \{0,1\}$) to the $i$'th vertex of the graph and we connect all vertices of $P_1$ to the all vertices of $P_2$. Then again in an optimal solution path $P_1$ is on one side and path $P_2$ on the other side.