Difference between AlphaGo's policy network and value network
In brief each net has a different purpose as you mentioned:


The value network was used at the leaf nodes to reduce the depth of the tree search.
The policy network was used to reduce the breadth of the search from a node (guiding towards promising immediate actions).


In general, you can use value function methods to find an optimal policy or directly search in the policy space to optimize a parametrized policy function (of course there are pros and cons). You can use function approximators (e.g Deep Nets) in each case. I see that mainly you are confused about the policy net so I focus my answer into this.

The policy net was first:

trained to do the moves that most likely a human would do given a board state (so input is a board state and output is a histogram that shows the probability of each action given that state). The net can approximate the probability function underlying the mapping from states to actions. It is reasonable to think to start building your policy from available data after all. After supervised training using experts moves the policy net could play the game sufficient (although far from a Master's level). Simply, you attempted to capture the general pattern of action selection of professional players.

Then,

it was trained in games with opponent itself, in order to optimize the previous-learned policy. This time its weights were updated using the REINFORCE algorithm. By doing this, you update the net parameters towards maximization of expected reward. Eventually you have a net that not only selects the actions like a professional player but also towards winning the game (However it cannot plan!).

After this step, they approximated the value function of a bit more noisy version of the learned policy, by regression (input is the state board and target the result of the game). You can use this network to affect the leaf node evaluation.

Conceptually speaking, the policy net gives you a probability over actions, but this doesn't indicate that you will end up in a good, for winning the game, state. AlphaGo had some "blind spots" and during the tournament did some really bad moves but also one exceptional move that a human could never had thought.

Finally you can use your planning algorithm (MCTS) in combination with these nets. Why we took all these steps? Briefly, the simple MCTS without any "intuition" would have failed.