With the winter here, the temperature dropping rapidly and snow and ice on the roads, it's not just interesting to consider the impact of all this on your (hybrid) electric vehicle, it is also very sensible! There are some things that remain the same when comparing with the conventional cars, but other aspects change.
There are a few key things to consider when driving in winter; the temperature and the grip on the road. As the temperature goes down, the battery gets colder too and similar as with the conventional car, this can be a problem. Most car manufacturers have implemented a battery management system, which is designed to control the battery temperature. While most attention goes to this system cooling the battery when it is generating heat by itself, it can also be used to keep the batttery at a comfortable temperature when it is freezing outside. This preserves battery life and keeps the battery in a good condition. This would ideally be done when the vehicle is parked and plugged into the grid, saving the power consumption from the battery itself.
The temperature also has an impact on the range the vehicle can travel. If the battery gets colder, the chemical reactions in the battery become slower, which decreases the current that can be produced at these temperatures. At some point the amount of current that can be generated can fall short of the demand, which a driver notices as a decrease in performance (less power available for acceleration for example).
The temperature also cools you down, when driving the car. This poses a few problems in itself; with the internal combustion engine there is a lot of excess heat available from the inefficient engine to heat up a lot of air and route that into the passenger cabin. In the case of a more electrified drive train, this excess load of heat is not available and alternatives have to be found to keep a warm and comfortable driving experience. One solution is to use an electric heater to warm up the air inside the car, but this has a serious impact on the power consumtion (typically around 3kW when switched on) and this decreases the range you can drive. An alternative is to use a petrol/gas powered heater, but for an electric vehicle this would not make a lot of sense to fuel up and burn fossils to get warm again; for most people driving an electric vehicle is also about not having to visit the gas station anymore. Another solution is in heating the passenger seats itself. Typically this is a heater that uses just 60-80 Watts of power, something that is having a lot less impact on your driving range. It is also more efficient, as it directly warms up the passenger, can be triggered by a person sitting in the chair (so it does not heat if there is no person on it for example) and it does not heat up the entire space in the vehicle. A last alternative I've only heard about so far is infrared heaters that are directed at the passenger. This would also eliminate a large load that just heats up all the air in the vehicle and target the passenger directly. Unfortunately I've not been able to find any actual applications of this solution in practice (yet).
Lastly, there is the problem of grip. Now this is a problem also applicable for the conventional vehicle, but since most hybrids or EVs come with low resistance tires, this can be a bigger problem for those to drive in winter conditions. Low resistance is good for getting more range in good driving conditions, but it also implies having less grip on the road when you need it most: with snow and ice. Take care that during such conditions you swap your low resistance tires for something that is more suitable for the winter conditions, but keep in mind that the winter tires have more resistance and thus decrease the range you can drive with them.
All in all winter conditions can be challenging for the (hybrid) electric vehicle, but you should not have bigger problems than the conventional car. The most common problem of an internal combustion engine is not a problem for the electric vehicle; the engine that won't start. The battery in the (hybrid) electric vehicle is usually kept in a better condition, mostly because it plays a more important role in these vehicles.