Does a cell phone charge faster while powered down compared to while being on?

The only thing that slows down a charge while the phone is on is when you're using it at the same time because the AC is then used to power the phone instead of charging the battery. The charging resumes the moment you stop using it.

When the phone is powered down, there is no immediate drain on the battery compared to when it is turned on. However, the difference in charge time may not be noticeable.

I charged my Nokia e62 and it took less than 20 minutes, and it was ON, while sometimes it takes 45 while OFF.

So no, cell phones don't charge faster while powered down; I think they do this so that you can actually use your phone while charging.

i charge mine on, it just saves turning it on and off, mine still charges in about 30 minutes, i have a razr

I believe it does, but probably depends on if you usually let your battery totally drain before recharging.