You now have two images, each made from identical RGB image data which look very different. One was switched to Lab mode so that your system can simulate the display of two images that have different palettes.
Each time you switch from Gamma = 1.0 to Gamma = 2.5 you are changing the "File Gamma" that Photoshop expects a file to have. A higher gamma means that less correction is required of Photoshop and lower numbers means that Photoshop will correct more. Two images with the same data, but different gamma values should be corrected differently, but Photoshop can't do two different gamma corrections simultaneously.
The colors in the palette of each image are not the same. It is impossible to see the two images the way they were designed to be viewed at the same time. And it is impossible to see the two images the way they were designed to be viewed using the palette from the other image.
The key to understanding this is to realize that RGB is not an objective color space. It is device dependent which means that it is a function of gamma, a white reference point, the phosphors in your monitor and other things. Repeat the experiment and watch closely when you first switch to Lab mode and notice that the Image One grey scale changes, but the rgb data remains the same. This is where it hits you that rgb data is really meaningless and that true reproduction requires gamma encoding in the image file, or monitors that have L,a,b guns, instead of R, G and B guns.