Winter camping is not for the beginner. There are few resources, and if things are tight, and lots of people trying to make it, then the worst possible situation. Most beginners would make a series of poor decisions and die. In the event of a SHTF scenario where people had to bug out in Winter, then unless you make intentional decisions and respond strategically then you'll die even with a very high skill level.
Your camp gear will be similar, as many of your tasks will be similar, but the addition of a good shovel specifically for use in shoveling snow would make many tasks easier. As snow is a pretty decent insulator, as long as you can pile it up to surround your tent, or can make a shelter from it, but it takes a huge amount of energy to do this. If you don't have food and water, it won't much matter.
Here are several videos. They're various adaptations of making it in the wild in winter. Obviously if you can find a place indoors, or much better underground, then it means modifying what you have to make camping work for you.
The first video shows using a common hammock tent. These can usually be found in Army surplus stores. They're small and portable, but meant for one person based upon the amount of weight they support. If you lashed your hammock fairly low and braced it off the ground and put snow around it, it would protect you from wind and insulate you more than merely dangling it in the air
A way of making it is to build directly in the snow using a T shaped structure. The trough in the center acts to pull the cold down into it and this makes the elevated sleeping area warmer. While igloos can be made, they're highly technical since none of the blocks is the same size, but fitted expertly and not plausible for a beginner. It is fairly easy to carve out a T shaped structure if there is sufficient snow, but you need a minimum of a shovel to do it efficiently. You must make an air hole. In areas where the snow is melting, your structure could collapse on itself and burying you in the process.
A 3rd option is a standard tent or tarp. If you have this much equipment, then making a hole and a vent tube is dealt with here. Obviously his mistake was making a new vent tube and not heating it first outside the tent (he got smoke that way). As it takes time to build a shelter, and often people travel during the day, camp is often built at the end of the day. If you're doing that, you sure better know how to set it up first, as it could take far longer than you expect with a dropping temperature. Imagine if you had a shelter smoke up when you were tired and cold and had to huddle outside. It would be just miserable.
Never sleep directly on the ground. As the temperature falls, and based upon your sleeping gear, you could lose consciousness and freeze to death.
I hope by looking at some of these, you're thinking how you would construct these kinds of shelters if you had to, what equipment and skills would you need, and how heavy they would be, and how much time would it realistically take to construct them, and how to disguise them so they would not be very obvious. If you're bugging out in Winter and trying to blend it, a structure like these can easily be seen.
If you had the 3rd setup, you'd be blessed with a lot of items and a much higher chance of surviving the night, and much more comfortably, but with the most chance of detection. If the tarp were white, then on first glance at a short distance, many people wouldn't notice you.