Variable Speed Drives vary voltage , Variable Frequency Drives vary pulse width.
no.... not quite.
a VSD just refers to a drive that changes the speed of a motor. if you are using a VSD in a dc application then it would vary voltage, yes. however a VSD in an AC application is a VFD. it's the same thing. VSD is just a name, it doesn't refer to anything specifically.
if you vary voltage on an AC motor it doesn't change the speed, it just changes the torque.
if you vary pulse widths on a VFD you are only changing voltage, this doesn't change speed either, only torque.
the output from a VFD is a PWM where the pulse width determines output voltage, and the rate at which the pulses widen, then narrow, then invert, then widen, then narrow, then invert is the frequency. for each positive and negative portion of the sinwave there are a number of pulses, starting very narrow and working wider until they narrow back down again, the widths of these pulses are determined by the commanded output voltage of the drive. higher voltage = wider pulses on average, though they still ramp up and down in width. each of these pulses is full bus voltage (in a single level drive). the inductive reactance of the motor windings smooth these pwms creating a sinusoidal current waveform in the motor. you can also use a reactor on the load side of the drive to create a sinusoidal current waveform. the carrier frequency determines how many of these pulses there are. the rate at which the pulses narrow and widen, then inverse determines the frequency and motor speed. the width of the pulses themselves only changes the voltage.
VFDs (and VSDs in AC applications) both vary frequency AND voltage to maintain constant torque in the motor. this can be done either by a fixed VF ratio (ie a 480v/60hz motor would be 8v/hz), or using vector control (open loop or flux vector using encoder feedback).