Powermac3,5?
That isn't an MDD, that is a Quicksilver.

Even so, that should still work as those models have a 4X AGP slot like the MDD.
And I am pretty sure that your X1950Pro card is 4X/8X compatible.

ALSO, with the G5's 8X AGP slot, that card should definitely have worked...if it is functional.
So I wouldn't recommend swapping the X1950Pro into the G4 unless you just want to confirm a problem with the card itself.

I assume you have a Radeon 9600, since you mentioned that card.
While most of those won't work in a G4, there a few variants that will (like the earliest Apple models that have an additional RF shield over the inside of the DVI ports and the PC/Mac compatible series).
But the chances of your card working in a G4?
Not good.

BTW - An R400 card will only work in your G4 if it is a flashed PC card (like the standard X800XT or the FireGL X3) as the ADC power connector is different on the G5 (than the G4).
And the Apple X800XT requires an AGP Pro slot that can provide more power than a standard AGP slot (PC R400s have a secondary power connector - once describe to me by Mark as a "PITA" to use in Macs, but it can be made to work).

But as flashed cards aren't recommended...
The PC Radeon X1950Pro AGP card is the first PC card to have official support when installed in Macs (and our genius developers got that working without an Apple rom).
If I had a G4 running right now, I'd double check for you, but don't doubt that Mark is completely correct when he says it should work if electrically compatible.

Since you know your current system is functional with the 9600 card, I recommend getting another video card for the G5.

An Apple Radeon X800XT makes as much sense as anything else as it natively supports AGP (while the R500 cards rely on a PCIe to AGP bridge chip), so it is just as fast or faster than the available R500 alternatives.
Sell the 9600, because the chances of it working in your G4 are minimal (and the few G4 compatible 9600Pro cards available are priced higher than a stock Apple 9600Pro or 9600XT).

[ Edited by Jim 21.12.2015 - 16:51 ]