Some common mistakes, definitely nothing close to exhaustive, so feel free to edit (I'll make it a community wiki if appropriate). If the point is about what you should do, the mistake is not doing it ;)
Underestimate the sleeping pad: You need a well insulated pad. There are various designs, but while R-Value isn't an absolute measure, it still is a good starting pointer: What is a good R value for a four-season sleeping pad?
With a too thin pad you will be miserably cold on the bottom (well all over with time) no matter how good your sleeping bag is.Forgetting to bring a second set of (under)clothing: Bring a second set of your baselayer clothing and store them completely watertight.
Have different gloves for different purposes and use them accordingly. Working gloves (mostly for digging the shelter) will get drenched eventually. Very warm gloves when you are "not doing anything", i.e. you don't need to touch snow (well, not more than occasionally) and you don't sweat. They are your backup gloves and should never ever get wet. Then I usually bring even a third pair that is used in between: I am not building stuff, but maybe cooking, walking, ... - they might get a bit wet, but not like when working.
Not using layering for clothing. I mean you should always do this outdoors, but in the cold it's just a necessity. You will need to adjust your clothing often due to activity level and/or temp/wind changes. You can't get cold but you also can't sweat, because you will get cold - so adjust early and often.
Forgetting to check their cooking gear at such conditions. Gas might not work or very badly at such low temps. I even had issues once lighting the gasoline when preheating the stove at ~-20degC. There's solutions, that's not really the point I want to make here, the point is, you don't want to find out whether it works when you are hours away from a warm place in the dark.