Try using Earth Mover's Distance (http://ai.stanford.edu/~rubner/papers/rubnerIjcv00.pdf). It measures how much it "costs" to optimally reshape one histogram into another, where an elementary transform (using the terminology of edit distance-like distance measures) is moving a unit of mass from a bin to a bin. If you have an implementation of EMD, then your distance is dist = EMD(H1, H2, D), where
H1 = [ 14, 13, 40, 31 ]; -- histogram #1 (first image)
H2 = [ 82, 8, 7 ]; -- histogram #2 (second image)
D = [ D11, D12, D13; ... ; D41, D42, D43 ]; -- (cross-bin) ground distance
Dij is a distance from the i'th bin of the first histogram to the j'th bin of the second histogram. For example, D21 is a distance between 'White' and 'Pink'. It may be defined differently, based on which color space you want to use, and further, based on how you define what it means for two colors to be similar. If I use RGB, and I do not care much as to how human eye perceives colors, then D21 can be the \ell_2 norm of the difference [1, 1, 1] - [1, 0.05, 0.7]. (As far as I remember, the paper I mentioned uses LAB instead of RGB.)
P.S. Besides color, you may include spatial information into comparison. EMD can handle it well, as long as you properly combine similarity of colors with spatial similarity in the definition of ground distance D.