Backpacking sleeping pads come in two basic types, foam and air, with some of the air pads being self-inflating.
What are the pros and cons of each and how would one decide which type to get?
Pads have several measurable characteristics that may be important for you when you make a decision:
Then the biggest difference is that air pads can be pierced, which may require a repair kit and knowing how to use it and being in conditions where you can use it. (Not sure the glue works in temperatures of -20)
So look at what you need in terms of all those characteristics (Typically, start with the R-value because that's one thing you cannot really compromise much on), then compare the pads that would work for you, and pick the right one
Cheaper (It's not long until they're riddled with puncture holes from crampons, rocks and sticks, or covered with stains from the sap and dirt, or have chunks taken out of them from getting snagged on stuff on the trail, getting stepped on, chewed on, etc.)
Less Comfortable (Less cushion, more firm, less soft.)
Less Warm (Thinner means less insulation.)
Bulky (Can only really be packed on outside of backpack.)
The best sleep system is a combination of both a foam pad and an air-mattress. The foam pad protects the air-mattress from the ground, moisture, and provides extra insulation, as well as a bit of firmness when you roll onto your side and you shoulder or hip fully compress your air mattress. It stops your air mattress from sliding around on your tent floor, and gives you something to kneel on when packing everything else up inside your tent.
The con to this system is of course extra weight, it's up to each individual to determine what they need to sleep comfortably, different styles of mattress and different combinations of sleep systems will suit different people.
Foam pad(s) for me!
Because: inexpensive, different types can be combined, always ready, waterproof, lightweight.
After years of trying just gave up on inflatables (or should we call them puncturables?). No matter how careful I was, they puncture, somehow. And the price! And yes, if you are trekking Everest (again, not) you will need both types ;)
Now when I travel I purchase pads in-country and leave them behind for others.
I used a three quarter length foam pad -- just the basic $20 (or so) Ensolite pad -- for decades until, growing older and more effete, I thought an inflatable pad, aka air mattress, would be more comfortable.
The inflatable air mattress -- more than $100 -- was no more comfortable, but was a PIA to deflate and inflate every time we moved camp. And, as @StrongBad mentioned in his comment, it was slippery: (a) I slipped on it, and (b) it slipped on the ground cloth. (Weather good -- no tent.) As a result of (b) it was punctured before the end of a 10-day trip. Moreover, even when we didn't move camp, we had to deflate the #$@& thing, because it could not be simply rolled up and tied to the pack as could a foam pad. Although this would not have been a nuisance if we had erected the tent, we didn't and it was.
In summary, although the inflatable air mattress might have been a bit lighter than the foam pad (I really don't know), it was an object that had to be pampered, and offered no extra comfort.
The point about comfort is highly subjective. If you are uncomfortable with the basic foam pad, an air mattress may be worth the hassle and expense.
Fully inflatable mattresses don't provide a great deal of insulation from the ground and tend to be fairly heavy plus they take a bit of effort to inflate and deflate which can become a nuisance if you are moving camp every day, especially in bad weather.
A closed cell foam pad is inexpensive, rugged and provides good insulation for light weight. Their main downsides are that they don't compress much and are rather bulky and take up a significant amount of space. Also being fairly thin and firm they don't do much for comfort on hard or lumpy surfaces.
The self inflating mats are a lot more compact than closed cell ones for a similar levels of performance while still being fairly easy to set up. They also tend to be a bit softer and thicker for the same weight and so may feel more comfortable, plus you can get extra thick ones, albeit with a corresponding increase in weight and bulk. Like fully inflatable ones they can be punctured.
Overall inflatable air mattresses aren't that well suited to backpacking. The choice between self-inflating and closed cell option really comes down to a compromise between price, durability and pack size which ultimately depends on your own personal priorities and the conditions you expect to encounter.
If you are unsure then a a basic closed cell foam mat is a pretty safe bet
You might also consider a Z-Rest, which greatly improves on the warmth of a foam mat. It also solves the problem of condensation under your mat, which afflicts both air and foam mats, by the "egg-crate" design allowing some airflow under the mat. Like other foam mats though, it is still bulky.