As others have already said, it shouldn't make much difference if both cards are fairly neutral. The use of gray cards has traditionally been used by still photographers to set exposure, and they really offer no advantage at all to white balancing. None.

The nice thing about owning a PDW-F800 is that it can tell me the exact color temperature when I white balance, instead of rounding off to the nearest 100K. So I just did some testing with five different white references. I tried to be very careful about using them in the exact same position with auto-iris turned on so the exposure would be correct. Here's what I got:

1) Brand new genuine Kodak gray card that has been stored in a light-proof envelope.
Color Temp: 2695K

2) The backside of the Kodak gray card that looks pure white.
Color Temp: 2723

3) A sheet of white Staples brand laser paper (98 bright / 28lb)
Color Temp: 2752

4) A white 5x8 unruled Staples brand index card
Color Temp: 2725

5) The white card from a set of Vortex Media's WarmCards.
Color Temp: 2683

As you can see, there is only a 69 degree range between the highest and lowest. I doubt that anyone can see that tiny little difference in the raw footage -- let alone in an edited piece. I can't hardly see it when I switch between A and B memories.

Notice that there is only a 12 degree difference between the fragile, uncoated Kodak gray card (considered to be the standard by which other references are judged) and the rugged, virtually indestructible white card that comes with a set of WarmCards. We spent a lot of time when WarmCards were being developed to come as close as possible to the neutral response of the Kodak card. +/- 20 degrees is farily common.

It's interesting that the backside (white) of the Kodak card is 28 degrees different than the front. But really, who cares? Small differences like these are practically impossible to see on-screen. The bottom line is that you can pretty much white balance off of whatever you want if you only want a plain vanilla white balance -- which with all the Sony cameras usually looks too cool.

Warning: shameless plug coming up . . .

For a better white balance (especially for interviews and head shots) try using WarmCards. WarmCards - White Balance Reference System

I use the Whibal for my photo work as I shoot raw and really like the workflow by taking a shot with the card in the scene and adjusting in post. Easy and spot on.

I even took the Whibal card to Sherwin Williams and had them scan it to make a color mix for the paint on my walls. It is a true neutral.

I tried this card with video and I have to say, the white cards gave a more true white balance.

So in my expereince, grey for photo, white for video.

I've seen many many posts on this forum about white balance and exactly what to use to balance with and they almost always seem exceedingly nit picky to me.

I am very careful and caring about the quality of my color and frankly I pick up the nearest white piece of paper that looks neutral or even a little cold if possible ( white paper with blue lines works well) and it always seems to work fine. But really napkins can do the job in a pinch also. Its mostly important to get in the ball park. Most techs I know use a chip chart but then they paint from there anyway.

Often balancing through 1/8 - 1/4 blue and 1/8 green gives a nice warm balance if you are under flourescents or worried about greenish cold balance.

The truth is that if I am in a controlled situation I always have a good monitor and I adjust color to that either in the pp menus or by cheating my color balance with gels. In an uncontrolled situation the light is usually mixed and the paper color is the least of your worries. Where you point the paper is more important and then understanding how the light on the paper compares to what your subject is.

That's my 2 cents.

I use a Vortex white card. I used to have a DSC white card, sadly lost it.

A long time ago I was in a training group for Z1s, where we ran a little test white balancing on a 'white' wall, a bit of white photocopier paper, and a DSC White Card.

The white card was definitively the best colour balance; flicking between it and the white paper, there were subtleties of tone in the shadow areas that were missing in the paper-balance.

It's even more valuable now, with the EX1 and its lack of '+' and '-' Colour Temperature shortcuts. NB: As the EX asks for the exposure of the white card to be dialled down to about 60%, I guess that makes a white card into a de facto grey card.

I shoot a lot in mixed tungsten/daylight/fluoro situations, and habitually 'wear' a set of Vortex cards; the white is used 95% of the time, but can cheat it with the warm and cool cards. I wish there were a gentler cool card though. Doug?

Here is a White/Grey collapsible card and has worked great for me.

http://www.bhphotovideo.com/c/product/375202-REG/Lastolite_LL_LR2050_EZYBalance_Grey_White_Card_.html
The camera focuses on it and white balances fast and accurate. In some situations it's easier to get WB off the grey.

If it works for you, go for it. But as you can see from my earlier post last night, you might as well us a piece ordinary paper if you all you want is a normal white balance.
White or gray, it's all the same to the camera.

Matt, thanks for the suggestion. If there are enough requests for a gentler cool card, maybe it might happen someday.