I wasn't around when they landed someone on the moon so I can't quite comment on that bit, but I can tell you what I (and the rest of my kind) use the extra processing power for:
Finite Element Analysis (simulating car crashes to make them safer before we crash the dummies in them).
Multibody Dynamics (Simulation of robot behavior saves a ton of money, we can simulate the different options before we build 10 different robots or spend a year figuring out something by trial and error)
Computational Fluid Dynamics (designing cars, jets and pretty much anything in between like windmills and how they affect their surroundings and how efficient they are)
Simulating Complex Systems (designing control schemes for anything from chemical plants, to cruise control to autopilots)
Computational Thermodynamics (Working on that tricky global warming thing, or just trying to figure out how to best model and work with various chemicals or proteins)
This is just the uses (that I know of) that more raw power can help out in Mechanical Engineering. I still have to wait about an hour for certain simulations or computations to run and they're not even all that complex yet. The faster these things run (even a few percent increases) can save us tons of time in the long run. And time is money...