In vanilla policy gradient, one plays the game to the end and then bumps the probability of all actions taken by the agent up (if AlphaGo won) or down (if it lost). This is very slow because there ~150 moves in an expert game, and we do not know which moves caused decisive victory or loss - i.e. the problem of "long term credit assignment". Also, it is actually preferable to compute the policy gradient with respect to the advantage of every action, so we encourage actions that were better than average and punish actions that were worse than average - otherwise the policy gradient estimator has high variance.

I think about MCTS in the following way: suppose you have a perfect "simulator" for some reinforcement learning task you are trying to accomplish (i.e. real-world robot grasping for a cup). Then instead of trying to grasp the cup over and over again, you can just try/"plan" in simulation until you arrive at a motion plan that picks up the cup.

MCTS is exactly a "planning" module, and it works so well in Go because the simulator fidelity is perfect. AlphaGo can't model adversary behavior perfectly, but MCTS and the policy network complement each other because the policy reduces the search space of MCTS. As long as the best adversary is not far away from the space that MCTS + policy is able to consider, AlphaGo can match or beat the adversary. Then, we train the value network to amortize the computation of the MTCS operator (via Bellman equality). Finally, self-play is an elegant solution for keeping adversary + policy close to each other.

For more rigorous mathematical intuition, Ferenc Huszar has a nice blog post on MCTS as a "policy improvement operator": http://www.inference.vc/alphago-zero-policy-improvement-and-...