Yes, for neural networks usually training them takes many orders of magnitude more resources than just using them.
For this particular example, training a system involves (1) analysis of every single game of professional go that has been digitally recorded; and (2) playing probably millions of games "against itself", both of which require far more computing power than just playing a single game.
I'm very aware of that. What I'm saying is that AlphaGo is not merely a neural net reporting best moves directly off the forward propagation. There are two nets which essentially act as proposal distributions for an exploration/exploitation tradeoff in the search space of game trees by which AlphaGo reads positions essentially out to the end of the game and ranks them by win rate (this is Monte Carlo Tree Search). The net moves are "nice" (I think they run at like 80% win rate against some other Go AIs? Maybe I'm misremembering) but the real heart of what makes AlphaGo play well is the MCTS which requires some vast resources to execute—live resources.
For this particular example, training a system involves (1) analysis of every single game of professional go that has been digitally recorded; and (2) playing probably millions of games "against itself", both of which require far more computing power than just playing a single game.