Is there any difference between calling for less power and sending less power?
Yes, there is a difference.

The circular thingy on the wall, called a dimmer, chops up the 60 Hz power sine wave in such a way that the overal apparent voltage available to the motor is less.

The knob on the fan causes different coils inside the fan motor to be switched in and out to trade off more torque versus lower power consumption in the motor.

Use the latter (the switch on the fan) to control its speed if you can.  The dimmer does some nasty things to the power line waveform.  This will not only make the fan run less efficiently, but also cause some radio interference.  The people that designed the fan added different taps to the coils inside the motor for a reason.  This allows still running the motor reasonably efficiently at the different speeds.  The downside is that you only get a small number of fixed speed settings, whereas the dimmer can control the fan speed smoothly.  However, it's a fan.  Three speeds should be good enough.