@Dave in Green
did a good job of explaining what is happening. The problem is not in the screen it is in your eyes when comparing A/B of Gray/White at the same time.
Your eyes each have an iris and it opens and closes to adjust to the average brightness of the image to allow you to get the best perceived contrast of any given image. Doing an A/B is asking your eyes to adjust two things at once and because the sample is likely small compared to the brighter screen your eyes are favoring the larger brighter image. then to compound the problem you can’t adjust your projectors brightness to two samples at once so you have it set to what is your normal brighter higher gain screen.
Something to think about and if you have a light meter you can try. If you are sitting say in your kitchen with the lights on reading the paper you will feel it is a nice bright room. Now if you go outside to read on your deck and the sun is shining you will say it is also a nice bright setting and might guess it is even twice as bright. If you take your light meter and measure it you might find out it is 10 times as bright. That is because your eyes in a split second adjusted trying to keep it constant.
Eye adjustment is measured in f-stops just like old manual cameras had adjustments. Each f-stop doubles the light or halves the light depending on the direction you are going. The human eye can adjust 26 f-stops I’m told. So that would be numerical in terms of times brighter 1, 2, 4, 8, 16, 32, 64, 128, ….. 26 times if you want to keep going you will see the fantastic range the eye has. There are some drawbacks. At night we can see the stars as our iris is wide open. Those same stars are still up there during the day but we can’t see them as the iris is closed many f-stops because of the sun and to see those faint stars we would need the ability the wide iris give us but if it was that open the sun would be blinding bright. So we get a huge range but not all at once.
Front projection can work over a variety of iris f-stops. By design the best is when it is on the very dark end of the range with a very good projector making very good black levels and of course in a very good room. That’s how CRT projectors worked. Now there is a trend to go much brighter combating ambient light forcing our eyes iris open more and giving us perceived blacks.
It’s all about the eyes. I have a .5 gain neutral gray stealth screen wall and trust me my perceived whites look nothing like gray whites. I have double the lumens I would need for a white 1.0 screen and the result is the same in terms of whites.