Color balance is, for starters, subjective. In terms of colors, it's very hard to reproduce a perfect representation of what is photographed. This is the case even with analog photography.

White balancing is a special (limited) case of color balancing where there's only a shift in what we call color temperature. The scale goes from yellow-ish through white to blue-ish. It is a monotone transformation (i.e. no tints will be swapped; they just move) and it is very subtle. Color temperature is different under different kinds of light (sun, bulb, TL...) and we don't really notice it because our brain corrects it, but cameras just pick up what they see.

Note that white balance is a necessary procedure. When decoding raw data (or developing film) you have separate channels which need to be mixed. Color balancing is inherent to mixing; not optional (well, you could just do nothing but than the image is unnaturally cyan and dark on my camera). Color balancing twice would just be like using a different algorithm once. (Basic algebra: g(f(x)) = h(x) if and only if h equals g after f).

Come to think of it, both images could have come from the same RAW data file, using different decoders. Passing around RAWs sounds like something professionals would do.