The simple answer is one takes care of business while the other takes care of art.  Sure, this is obvious, but I can't tell you how often I see the obvious getting overlooked.  The separation of duties can really be that simple, but there is one caveat I would present to you. 

There are those sticky little issues where business and art overlap (e.g., play selection vs. marketability may be one of those areas).  This is where an organization needs to decide what's more important and put one or the other at the top of the organizational hierarchy. You may have two people at the moment who get along terrifically and will always work out such differences, but that may not always be the case.  Organizations need to set policies and develop structures that are independent of current personalities (good or bad) and reflect the priorities of the organization.  

Not to be deliberately unhelpful here, but division of labor really depends on what you guys want.  There are many theatrical positions among which there is fluidity as to what is whose responsibility.  Just be sure that everyone is clear as to what is in whose purview.