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. 

**Changing clothes** is also important- 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.    

Also, **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).  

**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.

**Sun protection** is another common issue easy to overlook.  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.

Do not **cook in the tent.**  You could kill yourself with CO. 

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.