When you have both ski crampons and boot crampons, in what cases should either be used?
This is the type of ski crampon I have in mind. Thanks to @imsodin for providing the link to the picture.
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The answer to this question at its most basic level is generally use your ski crampons until you would feel more safe using boot crampons. I suspect since you asked this question you aren't very experienced with ski crampons and/or boot crampons so I'm going to talk a little bit about their use cases.
Ski crampons are only used when ascending, and used in conjunction with climbing skins. Boot crampons can be used when descending or ascending. Both boot and ski crampons are typically used when the snow conditions are hard. At a very basic level you can picture the climbing skin like the tread on the soles of your boots. Most of the time the skins/tread is sufficient to keep you from slipping but sometimes you want/need extra grip. This is where boot/ski crampons come into play. Crampons bite into the hard snow to give you more grip than your skins/tread alone. With boot crampons you basically have one type of crampon; the kind that straps to your boot. With ski crampons you have two kinds; dynamic and fixed. To quote my dad's website:
Since the dawn of ski crampons there has been an ongoing debate over dynamic versus static crampons. Dynamic crampons are crampons that rotate up with the forward motion of the ski and static crampons are crampons that remain fixed. Essentially this debate centers on the balance between ease of motion versus ease of activation. Without a crampon lock the crampon will rotate out of the snow as you lift your foot and kick forward. This reduces the amount of drag in the snow/ice as you are moving. If you engage a crampon lock the crampon is not free to rotate up so the user ends up pulling the crampon through the snow/ice. The benefit to this system is the user does not have to step down on the crampon with the sole of their foot in order for it to engage.
In other words dynamic crampons give greater forward efficiency for slightly less activation efficiency. This brings us to the heart of your question. If you ignore fun factor (which is a big factor I know), skis are used to increase your efficiency vs walking. This is a key determinate in whether you choose to use ski crampons or boot crampons. You should choose the crampons that you feel are safest and offer the most efficiency. Most of the time in my experience when skiing is an option ski crampons offer greater efficiency. In my opinion the efficiency spectrum is generally: skis without crampons --> skis with dynamic crampons --> skis with fixed crampons --> boot crampons. Of course your mileage may vary.
This is an excellent article comparing the two.
Use the ski when you need the grip but still want the flotation of the ski. You are limited to how steep you can go with a ski. This is a picture of people using the ski crampon.
The boot is better when it is too steep for ski, or the snow is so hard you need more bite.
You need to make sure you have boots compatible with both, but there are many boots and boot crampons like that.
A third option would be add skins. I pretty much recommend skins.
Ski crampons are used on frozen crust, most often in springtime when the crust will melt into corn snow before the descent. They cannot be used on extremely steep terrain (over 25-30 degrees) because getting purchase requires rolling the boots out so the skis are perpendicular to the snow, and beyond a certain steepness this is not comfortable or doable.
Boot crampons can be used on frozen crust, but also work well on hard snow, ice, and even rock. They are used once the slope is too steep for skinning.
Generally, ski crampons are used in spring where morning crust is an issue. I almost never bring them in winter conditions.
Boot crampons are used when you expect steep terrain where kicking steps with a boot would be difficult. I will almost always bring boot crampons when I bring ski crampons, but I often bring boot crampons without ski crampons.
@paparazzo Skiing down after climbing with ski crampons can be very easy if the crust melts into corn. It is not at all a Doug Coomb type adventure.