Sigh... Oh well...
Speed of an AC motor is ALWAYS determined by the supply frequency and the number of poles in the motor winding.
Changing the voltage without changing the frequency reduces the TORQUE of an AC motor by the square of the voltage. So for instance at 60% voltage, the torque is .6 x .6 = 36% of the normal torque. If you have a fixed load, the speed reduces because the motor now has insufficient torque to keep it spinning at full speed. DEPENDING ON THE MOTOR DESIGN, that may or may not work out too well, because MOST AC motors will try to keep running the same speed and pull more current to do it, until the motor burns up.
A variac or rheostat (a.k.a. dimmer) can change the speed on SOME TYPES of motors because they have a type of design that inherently limits the current flow, thus allowing the lower torque output to occur without burning themselves up. These are called Shaded Pole motors and most likely your fan is one of these if it hasn't burned up from your test. Another type is a "Universal Motor" which is really more of a DC motor to which you supply AC. This is what you find in small appliances and portable tools, not likely on a fan.
A VFD changes the voltage AND frequency together at a predetermined ratio, which allows the motor to maintain a constant torque while varying the speed by keeping the ratio the same as it was at full speed and full voltage. But it does so by converting the AC to DC, then using high speed transistors to fire DC pulses in what is called a PWM pattern to the motor to recreate a "pseudo sine wave" that the motor reacts to AS IF it were AC. This works great on 3 phase motors. The problem with most types of single phase motors is that the pulsed DC is incompatible with the design of the motors, and you can damage the motor, the VFD or both. Usually both.
Now go back to what I said earlier. Any further discussion is pointless without knowing exactly which kind of motor you have. If it plugs into the wall, and we ASS-u-me you are in North America somewhere (this is an international forum), then the chances are about 99.9% it is a 1 phase motor. That doesn't tell us enough however.