I've always been on the edge of switching from regular flex in all my clubs to stiff. Different fitters have always said that based on my swing speed and distance I would be a good match for going stiff but I've always found a few issues:
Going stiff definitely causes me to lose feel with the clubs and the trajectory definitely is lower with stiffer shafts. I've been told that the flipside, shafts being not flexible enough, will cause someone to really balloon the ball.
What i've been told and found a little myself is that going too stiff can cause you to actually accentuate a slice or hook as the shaft isn't flexing enough and doesn't have enough time to square up properly. Think about it, if the shaft is flexing it will lag behind a bit and give you perhaps a fraction more time to help square up the clubface. I think this is probably mostly true for higher handicappers or slower/smoother swingers.
I work in a sports store and we constantly see people go too stiff with the shaft when they don't have the swing speed to really get the benefit out of it. If you see what a lot of golf teachers say that their best recommendation for most players is to have more loft and a more flexible shaft in their drivers.
Just figured I'd put my two cents in as i think a lot of people associate stiff with straight when they may not have the swing speed or skill to make that the case.