The images look pretty similar, but your eye can tell the difference, specially if you put one in place of the other:
For example, you can note that the flowers in the background look brighter in the averaging conversion.
It is not that there is anything intrinsically “bad” about averaging the three channels. The reason for that formula is that we do not perceive red, green and blue equally, so their contributions to the intensities in a grayscale image shouldn’t be the same; since we perceive green more intensely, green pixels should look brighter on grayscale. However, as commented by Mark there is no unique perfect conversion to grayscale, since we see in color, and in any case everyone’s vision is slightly different, so any formula will just try to make an approximation so pixel intensities feel “right” for most people.