In today's economic environment, one of the major ways in which state governments come into conflict with one another is in their competition to attract businesses to their states. This competition (as seen in the New York Times article in the link below) often takes the form of trying to attract business from one state to relocate in another.
This sort of conflict is possible because states are free to set up their own tax systems. With the economy stagnant, they often try to give tax breaks to lure businesses to their states. When this happens, state governments can get into bidding wars that end up hurting their states.
This is a major form of conflict between state governments today.