I bought Noodler's Whiteness of the Whale (WotW) because I was not sure whether it would make a difference in trying to mix colors, but read some indications suggesting it would. Below are the results of an experiment to try to answer the question.
Strathmore Writing (25% cotton, natural white, wove, 24lb):
Georgia Pacific Multipurpose Paper (20 lb, bright white, Walmart):
Unless otherwise indicated the writing is from a dipped Jinhao X750.
I learned more definitively the things I did wrong in the above and things I should have done differently than actually answering the question.
From the above my current inclination is that the answer is either "inconclusive" or "no". The inks definitely behave very differently: the WotW mix is "chalky" and does not shade as much and ink diluted with water to that degree has too much surface tension to be consistent when dipping as I was. With respect to color I am not sure there is any difference that cannot be explained by the different behavior of the ink. There is some slight variation in color (more visible in the scan I think), but it is such that I am not sure I can say it is significant given the other sources or error in my method.
I think I will need to actually load them in a FP and write with them for a time to see if I learn anything more. I suspect one of the big behavior differences to be when the ink starts to dry on the nib, I know the water dilution will darken; but I suspect that will be less the case with the WotW mix. Though I think the WotW mix would dry-out/clog more quickly in the same situation.
Currently I am using a pen loaded with the 1:20 Distilled water mixture after adding some Photo Flo to bring the surface tension down to a usable level. I maybe will make another post on that after a time because amazingly the ink is rather usable after being diluted 20x (!) and the color looks to me to be very similar to Diamine Beau Blue from this Month's ink drop (probably a little less red than Beau Blue).
I thought i was going to try this experiment with a few inks to show the various results, but after not finding anything interesting here and knowing how much time this took probably won't bother.
The WotW mix definitely looks cooler though, especially if you will be putting it in a demonstrator:
Hopefully this is helpful to someone. I would be interested in knowing whether others have experience that corroborates or refutes my findings.