Why don't different output weights break symmetry?
It is fine if a hidden node has identical initial weights with nodes in a different layer, which is what I assume you mean by output weights. The problem with weight-symmetry arises when nodes within the same layer that are connected to the same inputs with the same activation function are initialized identically. 

To see this, the output of a node $i$ within a hidden layer is given by
$$\alpha_i = \sigma(W_i^{T}x + b) $$
where $\sigma$ is the activation function, $W$ is the weight matrix, $x$ is input, $b$ is bias. 

If the weights $W_{i}=W_{j}$ are identical for nodes $i,j$ (note that bias is typically initialized to 0), then $\alpha_i = \alpha_j$ and the backpropagation pass will update both nodes identically.