Convolution v.s. Cross Correlation in Digital Image Processing
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.