Disadvantages?
Well, in general, based on my experiences, which may not be technically correct, but since my heart/soul/mind/feel/touch says otherwise, the MAIN change in headroom does not necessarily mean its better.
Higher headroom means less distortion.
Which can work if you run super hot pickups into say... an MXR phase 90.
Running at 9v, the phase 90 is a dirt box with phasing on it. Really. Quite musical I think. Running at 12v (the phase 90 can't do 18v I think), the higher headroom = a cleaner, proper phaser.
But then, running low output pickups with the phase 90 which does not over push it, having 12v kinda makes the phase 90 too "clean" and it loses that dirty sweet phasing it is known for.
Get the drift?
Also, there are usually changes in the 'size' of the sound. Usually tends to sound BIGGER, more open, like things are more spaced out.
Finally, you might encounter higher noise. I know that a CE-2 and Phase 90, at 12v, the ticking/swooshing gets louder. On pedals that are made for higher voltages, shouldn't be an issue.
And 9v/18v for emgs... that's another can of worms.