Convolution v.s. Cross Correlation in Digital Image Processing I understand that from mathematical point of view, only difference between Convolution and Cross Correlation is that Convolution is commutative, while Cross Correlation is not. 

In many articles Cross Correlation is explained as measuring similarity between two signals, and Convolution is explained as calculating relationship between two signals.

My question is this, what is the point of commutative property? How does it make Convolution different from Cross Correlation in practice? Or is it just there for "better" mathematics and easier proofs?
Apart from the formulae, let us go back to their actual meaning, and how they are derived. Talking about convolution: this operation is inherent to Linear-Time-Invariant (LTI) systems. In other words: if 


you want to analyse a system that is linear, and time-invariant, 
or you want to apply a processing or a filter that does not vary in time or space


then the convolution is the ONLY suitable operation. 

While cross-correlation is a measure of similarity between two series, computed as a function of the displacement of one relative to the other. Many choices are plausible, the most common is "bilinear" and compatible with energy preservation.