Why does the spa pump hum but not rotate

Another solution might be to just buy and install a new pump. If that solves it, you're golden. If not, you now have a spare for when the pump does eventually die.

Obvious caution: Electricity and water. Be VERY sure of what you're doing when you make the connections and seal them up.

(I presume you've already checked the obvious -- made sure all valves are fully open, that any flexible pipes aren't kinked, and so on.)

After pulling the motor out (which required draining the spa and disconnecting the PVC connections) and removing the cover, I found another capacitor. Googling the part number indicates that this is a start capacitor. It helps the motor start.

The motor also has a capacitor on the outside, and it's the run capacitor. That's the one that was replaced first.

After replacing the start capacitor, the motor started and is running.

http://en.wikipedia.org/wiki/Motor_capacitor

It turns out that Spa/hot rub AC motors like this have both a start and run capacitor, as do blowers on forced air furnaces and air conditioning units. If an AC motor like this won't start, there's a good chance it's the start capacitor. The start capacitors tend to fail eventually, the run capacitors last much longer, apparently.

My spa motor

Start capacitor

Run capacitor