Corresponding authorship is often used to signal the person who is the primary contributor to the work. I've seen this in fields like economics where there is a relatively strong norm of listing authors alphabetically. In these cases, the primary contributor is signaled with the corresponding author designation. I've seen people go so far as to list corresponding authorship on their CVs when they list their papers although I think this is far from the norm.
In fields where authorship is not simply alphabetic, I've seen the corresponding author designation used to flag the second author as a major contributor because, even in cases of so called "dual-first authorship," somebody's name has to come first.
In lots of other fields, it caries little signal value and really is just used to mark the person who is in the best position to answer questions about the work. Of course, the person marked as best able to answer question will, mostly likely, be someone who was involved in a wide variety of the research reported in the paper.