Getting those little stress fractures when running makes sense because your feet end up supporting several times your body weight with each step. This just doesn't happen with riding.
Tangent, it could be any number of things, depending on what type of pain it is. How old are your shoes? If they're fairly old, they may have just broken down with use. (Though I imagine it would take far longer with bike shoes than with regular ones since you're not walking so much with them.) Your cleat placement may need to be tweaked. Or you may just need to switch to a stiffer shoe and/or road pedals. I've been contemplating making the switch to road pedals because I've been getting hot spots.