According to Deep mind's article, Alpha Zero searches roughly 10,000 moves per decision, while Stockfish is around 10,000,000 moves. I'm aware that Alpha Zero uses Monte Carlo Tree Search while Stockfish uses minimax, but why would this cause 1000 times less moves to be considered?
Very simple. Running a large neural network with floating numbers can’t be quicker than running a simple C++ function on integers.
AI machine learning is not new to chess. There were serious attempts before Google established. Unfortunately, nobody had the determination, skills and resources to do a good job. Nobody wanted to invest money. Before Google's chess journey, the chess engine community believed:
AI chess will not beat classical programming because neural networks (or other models) run much slower.
While I was in the middle of writing this question I came up with a probable answer, so I'll write it here. AlphaZero is actually running through the same positions far more than 10,000 times, it's just doing it on a set of positions roughly 10,000 large. So each position has many playouts run on it, each >= in length than the last.
Stockfish looks at millions of positions and evaluates each only once (not taking things like iterative deepening search into account). Meanwhile, AlphaZero repeatedly looks at promising branches based off their previous preformance. This explains why it only looks at 10,000 moves, since it's usually only sampling the best it can find.