Thank you. I didn't know that.
I found the answer. I was told to make sure the color profile was the same. I looked at it again but both files were sRGB IE61966 - 2, and they still are.
But I dugg a little deeper and I found it. So even though both profiles were the same, they really weren't. I had to go to Edit > Assign Profile > and select sRGB IE61966 - 2 ! Apparently, Mac (colors) was selected for the 'screenshot' file and that caused the problem. I think this happens by default.
Assigning a profile does change the colors of the 'screenshot' file. That's no good, because I wanted to use the color picker. The way to deal with this problem is to keep it like it is (do not assign profile) and duplicate / move the layer to the sRGB IE61966 - 2 file, and use the color picker in that file. Which is how I did it before I started questioning everything :-) I now understand why those colors weren't the same.