The motor you have is called (and I'm not sure about the actual phrase) capacitor start/capacitor run. One of the capacitors does exactly what you surmise, it gives a boost of electricity to the motor to overcome the load at startup. The other is there to help out during normal operation.

If you can, disconnect the motor from the pump and see if it starts up without an issue.

Also, if I'm not mistaken, there will be a centrifugal switch somewhere on the back side of the motor. When the motor spins down at shutoff, this switch engages, allowing the power from the capacitor to reach the windings of the motor at startup. When this device fails, the motor doesn't get the boost it requires to start up, and you generally just get a humming sound (and possibly a motor that will trip the breaker).

If you can get the motor to run (you can usually get a capacitor start motor to run without the aid of the capacitor by spinning the shaft and applying power before the motor stops spinning), listen for a tell-tale "click" when the motor winds down after shutoff. It should occur just before the shaft stops spinning.

I could be wrong about all of this - but most of the time the capacitors aren't the problem, either the aforementioned switch, or the windings of the motor itself.

Your safest and probably best course of action is to take the motor to a repair shop that specializes in things like this. You might be surprised at how cheaply the motor can be repaired.