To add to the already-good other contributions:
Wearing cotton clothing is a common mistake. When cotton gets wet (from sweat or external sources), it doesn't quickly get dry, and that cold moisture close to your body conducts heat away. To underscore another answer's point even more, layering is also important; use non-cotton layers wherever possible. Have at least one layer that's a good windbreak.
Being reluctant to change clothes is a common issue: First time winter campers might not want to do this because they may be already cold and think it'll be colder exposing their skin to that cold air for the short time needed to change, but the extra warmth from fresh dry clothes is impressive. You can also try changing inside a sleeping bag to mitigate the effect.
I've also found that people tend to overlook the warmth value of silk long underwear. It's not super cheap, but it's warm, compact, and lightweight, and the extra happiness you can get from being warm makes winter camping a lot nicer.
Failure to account for battery performance degradation at lower temperatures is a common error that leaves people without enough spare batteries for headlights etc. (which you use more as there are more hours of darkness).
Failure to stake out the tent for ventilation is a common error. It seems counterintuitive, but make sure you use those ropes that hold a gap between the rainfly and tent body to allow for ventilation (and to some degree, a layer of air insulation). You will produce water vapor in your breath at night and if that's allowed to build up you'll feel cold and clammy in the morning.
Not having enough traction underfoot is another common error. You can buy YakTrax, ice spikes (which are similar to crampons but many fewer spikes and less sharp), or crampons. Good ankle support can help avoid rolled ankles.
Dehydration is a common issue because the air is drier and you need to drink more to stay hydrated, especially when doing active outdoor physical activities. Even basic camping (setting up the tent, fire-building, cooking, etc.) uses more physical exertion than you might do at home. Drink plenty of liquids, and include some salts/electrolytes in the mix.
Overlooking sun protection is another common issue. In the winter you have not only the sun from directly above, but reflecting from the snow. Have good eye protection and consider still using sunscreen.
Cooking in the tent can kill you with CO (or fire). Don't do it.
Do consider putting a bottle of hot water in the sleeping bag with you at night. If you don't have hot water, at least liquid water. Then it's more likely to stay liquid until the morning. Don't fill water bottles completely if they might freeze; leave about a third of the bottle empty for expansion that doesn't bust the bottle. If you have options, choose insulated-wall bottles instead of single-wall plastic.
If you went unprepared and are freezing in wet clothes, and think you might be suffering from hypothermia, avoid going to sleep and if you reach that later stage when you feel much warmer, don't take off your clothes as a result.
In general, I think a common mistake is underestimating how much fun and how much of a positive learning/growth experience winter camping can be. Do it enough times and it also provides psychological immunity to long power/heat outages from winter storms at home.