The VFD can be of a smaller frame size in the higher voltages.
Other than that the actual control of a VFD is the same at either voltage.
( VFD's generate their waveforms by tapping a 'tank' of electropotential -- held up by a DC / rectifying circuit. This means that a VFD -- correctly sized -- can generate 3 phase power from a single phase source. This truth is ruining the market for rotary phase converters -- since the Chinese are now exporting VFDs at astonishingly low prices. Check out eBay. )
( When tapping a single phase power source, the VFD will be oversized -- quite a bit -- since it will have only the one wave to rectify. This is often still very much cheaper than a rotary phase machine -- which has to run, and run, and run... if you expect to have 3-phase power on tap. So, as a general rule, purchase a 15hp VFD to run a 5hp machine if you are limited to single phase power. )
The cost of adding a transformer in front of a VFD so that it can be downsized one frame size is uneconomic.
I rather suspect that the nostrum got started when a new generation 480V VFD was compared to an older generation 208V VFD.
ALL of the new stuff is now electrician friendly. That is: expect it to be able to link into multiple control schemes -- including building automation.
( HVAC is a HUGE user of VFDs these days. Blowers and compressors running at reduced speed mean astounding energy savings. So much so that when explained the normal civilian response is incredulity -- to the point of tuning you out. So don't ever get technical with the average Joe on this matter. )
( Fluid friction scales to the CUBE of the fluid speed: twice as fast requires eight times as much energy; four times as fast requires sixty-four times as much energy. The inverse, that cutting the speed by 75% reduces energy by 63/64ths is too much to comprehend. )
( It gets worse if you ever mention PWM or any other technical details. So don't. ) ( Pulse Width Modulation )