Maximize vertex cover weights with bounded edge weights in a connected subgraph
This is already NP-hard in the special case that the graph is a tree, since Knapsack can be reduced to such an instance of this problem: Let $v_i$ be the value of the $i$-th item, and $w_i$ its weight.  Create a vertex of weight $v_i$ for each item $i$, and a single extra vertex that is connected by an edge of weight $w_i$ to each item $i$.  A capacity-$b$ knapsack can be filled with items of value at least $k$ iff the solution to the constructed instance of your problem with total edge capacity $b$ can achieve a total vertex-value of $k$.