None at all. For JPEG of course. The WB camera setting is simply a metadata suggestion for a raw converter (and each will render the WB differently). All the camera settings for sharpness, picture styles, color space and WB play no role or affect the raw data.
I agree with this with one caveat if only shooting a gray card. The caveat being that if the actual WB is so far out from the set WB, depending on the camera, it is possible to end up with a clipped color channel when adjusting in raw. I've only seen this happen under extreme conditions and it's usually the red channel, but it does happen.
But the question involves using the color passport.. Instructions say shoot the grey card first, set your custom WB, and then shoot the color checker.
Are they doing this because the custom WB then provides the most accurate color checker capture for use in building your profile, or because they think this is an easier way to remember vs. counting on people to remember to adjust the color checker WB prior to building the profile.. or does it just work better?
I've done both because even though I shot the grey slab.. I then forgot to set the custom WB.. So I used the grey slab to set WB or the white squares.. then changed the WB accordingly on the color checker image.. then built the profile. Really, it's possible to just shoot the color checker and use the white squares on it to set WB, then run the profile builder.. but if this is a better way.. or makes no difference.. why did Xrite go to all the expense/trouble of including the grey slab and make their product more complicated than it needs to be?
My observation is it 'appears' to make a difference if you follow the directions. I get very accurate skin tones this way. Adjusting WB after the fact don't seem to be as accurate. Why? It could be the sequence, or it could be the way I set WB in LR..