Skier weight is a big factor in choosing ski boot stiffness (flex rating). If you weigh a lot, a soft boot will bottom out and over flex, so not perform well. If you are light, then you can use a soft boot or a stiff boot, its just personal preference. The best way to determine what you like is to ski a range of boot flexes to get the experience. Its not very practical, however, as boot demo's aren't done like ski demo's. You can try them on in a store, so you can get a "static" feel for them. You might want to bring a pair of skis with you to click in and then mimic ski motions and flex them forward and back. I have done that in my basement when I bought new boots during the summer and couldn't go ski them right away. I believe there is a range of flexes that will work well for a skier, and each skier will have a different range, again a personal preference thing. For me at 240 lbs, a 140-120 flex range will ski well. Softer than 120 is too soft and I over flex the boot.
The trade off's in performance between stiff versus soft boots? This can and will get really complicated and have lots of different ways to look at it. I know a boot is too soft for me when I am spring skiing and hit a patch of sticky wet snow that feels like someone just stomped on my brake pedal. If my boot is too soft, I lean too far forward and loose control of my skis. I go over the handlebars. Another indicator is when I flex forward to drive my ski tips and not much happens with the ski, the boot flexes instead of the ski edge really digging in. Too soft a boot leads to delayed/weak edging. Finally, if the boots are too soft the boots are flexing and distorting so much my feet are moving in the boot and this causes localized pain. If your feet are comfortable when standing in your boots, but they hurt during/after skiing then that indicates your boots are too soft (or too big, or too loose) and your foot is moving in the boot and that causes the forces from skiing to be concentrated in one area of your foot instead of being spread out over your whole foot and leg. This drastically increases the magnitude of the local force and starts to cause pain or it compresses the liner to the point your foot contacts the shell and causes contact pain. If the boot was stiffer and held your foot better, the force would not be concentrated and the liner would be cushioning your foot. My feet feel less pain in a stiffer, low distortion boot.
In a stiff boot, shin bang or shin pain can be an issue. You need good tongue cushioning and a good instep fit to keep them comfortable. Another issue is having them set properly for forward lean and cant angle. In theory, if they are set properly for your skiing stance, then you shouldn't need a lot of flex range of motion while skiing. If your boot has low forward lean and you like to crouch low while skiing, you will be constantly trying to flex the boot and that will be tiring and uncomfortable after a while. The opposite applies as well. The downside of stiff flex, is the range of motion is reduced and the ability to absorb impacts by flexing is a lot less. It is really important to have a stiff boot setup properly. This would include binding ramp angle, boot footboard ramp ange, forward lean setting, and cant/cuff angle. If its all set right for your body and style, a stiff boot will ski very comfortably. I have experienced both ends of the spectrum with stiff boots.
Lastly, I think the stiffer the boot, the higher its performance period. With the exception of mogul skiing and being able to absorb impacts by flexing the boot. I don't know this, but I am curious what flex rating Olympic caliber mogul skiers typically use? I would guess its around 120 or so. I did learn through my race coaching experience that WC racers use very high flex rated boots for slalom skiing (150-170) and softer flex boots for downhill speed events (?110-130?). I was told DH racers need more absorption in their boots so they use a softer flex.
Softer flexing (consumer) boots will be more absorbing of terrain impacts, be less precise, allow your foot to move more and possibly hurt your feet more when skiing hard. After a day of skiing my AT boots (130 flex) with light weight touring liners in the resort, my feet felt like they were worked over and pretty sore. No sharp pains, just overall sore. After a similar day in my 140 race boots, no soreness at all. I have since changed the liners in my AT boots to a Mimic liner which is thicker and holds my foot better. My foot soreness has completely away. So the liner made the difference for me. I guess its not all the boot, but the boot liner combination. Or its really about the fit and foot hold of the boot and then shell distortion comes into play on top of that.
@Jim Kenney what are your thoughts or focus ?