Spa/hot tub A/C pump hums, but does not run
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